GCSE Maths: Prime factorisation (Higher)

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 12 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Split the number into any two factors (a factor pair). Choose two numbers that multiply to give it. Any pair will do: you will always reach the same primes in the end.
   12 = 3 × 4
   4 = 2 × 2
2. Keep splitting every number that is not prime. Circle each prime when you reach it. A prime cannot be split any more, so that branch stops. Stop when every branch ends in a circled prime.
3. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   12 = 2 × 2 × 3
4. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   12 = 2² × 3

Check. Multiply the primes back together:
2 × 2 = 4
4 × 3 = 12
The result is 12, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 20 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   20 ÷ 2 = 10
   10 ÷ 2 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   20 = 2 × 2 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   20 = 2² × 5

Check. Multiply the primes back together:
2 × 2 = 4
4 × 5 = 20
The result is 20, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 28 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Split the number into any two factors (a factor pair). Choose two numbers that multiply to give it. Any pair will do: you will always reach the same primes in the end.
   28 = 4 × 7
   4 = 2 × 2
2. Keep splitting every number that is not prime. Circle each prime when you reach it. A prime cannot be split any more, so that branch stops. Stop when every branch ends in a circled prime.
3. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   28 = 2 × 2 × 7
4. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   28 = 2² × 7

Check. Multiply the primes back together:
2 × 2 = 4
4 × 7 = 28
The result is 28, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 45 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   45 ÷ 3 = 15
   15 ÷ 3 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   45 = 3 × 3 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   45 = 3² × 5

Check. Multiply the primes back together:
3 × 3 = 9
9 × 5 = 45
The result is 45, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 50 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Split the number into any two factors (a factor pair). Choose two numbers that multiply to give it. Any pair will do: you will always reach the same primes in the end.
   50 = 5 × 10
   10 = 2 × 5
2. Keep splitting every number that is not prime. Circle each prime when you reach it. A prime cannot be split any more, so that branch stops. Stop when every branch ends in a circled prime.
3. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   50 = 2 × 5 × 5
4. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   50 = 2 × 5²

Check. Multiply the primes back together:
2 × 5 = 10
10 × 5 = 50
The result is 50, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 63 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   63 ÷ 3 = 21
   21 ÷ 3 = 7
   7 ÷ 7 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   63 = 3 × 3 × 7
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   63 = 3² × 7

Check. Multiply the primes back together:
3 × 3 = 9
9 × 7 = 63
The result is 63, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 72 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   72 ÷ 2 = 36
   36 ÷ 2 = 18
   18 ÷ 2 = 9
   9 ÷ 3 = 3
   3 ÷ 3 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   72 = 2 × 2 × 2 × 3 × 3
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   72 = 2³ × 3²

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 3 = 24
24 × 3 = 72
The result is 72, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 90 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   90 ÷ 2 = 45
   45 ÷ 3 = 15
   15 ÷ 3 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   90 = 2 × 3 × 3 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   90 = 2 × 3² × 5

Check. Multiply the primes back together:
2 × 3 = 6
6 × 3 = 18
18 × 5 = 90
The result is 90, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 120 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   120 ÷ 2 = 60
   60 ÷ 2 = 30
   30 ÷ 2 = 15
   15 ÷ 3 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   120 = 2 × 2 × 2 × 3 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   120 = 2³ × 3 × 5

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 3 = 24
24 × 5 = 120
The result is 120, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 150 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   150 ÷ 2 = 75
   75 ÷ 3 = 25
   25 ÷ 5 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   150 = 2 × 3 × 5 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   150 = 2 × 3 × 5²

Check. Multiply the primes back together:
2 × 3 = 6
6 × 5 = 30
30 × 5 = 150
The result is 150, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 180 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   180 ÷ 2 = 90
   90 ÷ 2 = 45
   45 ÷ 3 = 15
   15 ÷ 3 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   180 = 2 × 2 × 3 × 3 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   180 = 2² × 3² × 5

Check. Multiply the primes back together:
2 × 2 = 4
4 × 3 = 12
12 × 3 = 36
36 × 5 = 180
The result is 180, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 200 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   200 ÷ 2 = 100
   100 ÷ 2 = 50
   50 ÷ 2 = 25
   25 ÷ 5 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   200 = 2 × 2 × 2 × 5 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   200 = 2³ × 5²

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 5 = 40
40 × 5 = 200
The result is 200, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 360 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   360 ÷ 2 = 180
   180 ÷ 2 = 90
   90 ÷ 2 = 45
   45 ÷ 3 = 15
   15 ÷ 3 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   360 = 2 × 2 × 2 × 3 × 3 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   360 = 2³ × 3² × 5

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 3 = 24
24 × 3 = 72
72 × 5 = 360
The result is 360, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 504 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   504 ÷ 2 = 252
   252 ÷ 2 = 126
   126 ÷ 2 = 63
   63 ÷ 3 = 21
   21 ÷ 3 = 7
   7 ÷ 7 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   504 = 2 × 2 × 2 × 3 × 3 × 7
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   504 = 2³ × 3² × 7

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 3 = 24
24 × 3 = 72
72 × 7 = 504
The result is 504, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 630 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   630 ÷ 2 = 315
   315 ÷ 3 = 105
   105 ÷ 3 = 35
   35 ÷ 5 = 7
   7 ÷ 7 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   630 = 2 × 3 × 3 × 5 × 7
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   630 = 2 × 3² × 5 × 7

Check. Multiply the primes back together:
2 × 3 = 6
6 × 3 = 18
18 × 5 = 90
90 × 7 = 630
The result is 630, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 1000 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   1000 ÷ 2 = 500
   500 ÷ 2 = 250
   250 ÷ 2 = 125
   125 ÷ 5 = 25
   25 ÷ 5 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   1000 = 2 × 2 × 2 × 5 × 5 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   1000 = 2³ × 5³

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 5 = 40
40 × 5 = 200
200 × 5 = 1000
The result is 1000, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 1800 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   1800 ÷ 2 = 900
   900 ÷ 2 = 450
   450 ÷ 2 = 225
   225 ÷ 3 = 75
   75 ÷ 3 = 25
   25 ÷ 5 = 5
   5 ÷ 5 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   1800 = 2 × 2 × 2 × 3 × 3 × 5 × 5
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   1800 = 2³ × 3² × 5²

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 3 = 24
24 × 3 = 72
72 × 5 = 360
360 × 5 = 1800
The result is 1800, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 2310 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   2310 ÷ 2 = 1155
   1155 ÷ 3 = 385
   385 ÷ 5 = 77
   77 ÷ 7 = 11
   11 ÷ 11 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   2310 = 2 × 3 × 5 × 7 × 11
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   2310 = 2 × 3 × 5 × 7 × 11

Check. Multiply the primes back together:
2 × 3 = 6
6 × 5 = 30
30 × 7 = 210
210 × 11 = 2310
The result is 2310, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Method. Evaluate each power first, then multiply.

1. Work out each power separately. A power means repeated multiplication of the base.
   2³ = 2 × 2 × 2 = 8
   5 = 5
2. Multiply the results together. The numbers in the product are multiplied.
   8 × 5 = 40
3. Write the answer. The value of the number.
   2³ × 5 = 40

Check. Prime factorise 40 again and you should get 2³ × 5. ✓

Watch out. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.

Method. Evaluate each power first, then multiply.

1. Work out each power separately. A power means repeated multiplication of the base.
   2² = 2 × 2 = 4
   3² = 3 × 3 = 9
2. Multiply the results together. The numbers in the product are multiplied.
   4 × 9 = 36
3. Write the answer. The value of the number.
   2² × 3² = 36

Check. Prime factorise 36 again and you should get 2² × 3². ✓

Watch out. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.

Method. Evaluate each power first, then multiply.

1. Work out each power separately. A power means repeated multiplication of the base.
   2 = 2
   3² = 3 × 3 = 9
   5 = 5
2. Multiply the results together. The numbers in the product are multiplied.
   2 × 9 = 18
   18 × 5 = 90
3. Write the answer. The value of the number.
   2 × 3² × 5 = 90

Check. Prime factorise 90 again and you should get 2 × 3² × 5. ✓

Watch out. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.

Method. Evaluate each power first, then multiply.

1. Work out each power separately. A power means repeated multiplication of the base.
   2⁴ = 2 × 2 × 2 × 2 = 16
   3 = 3
   7 = 7
2. Multiply the results together. The numbers in the product are multiplied.
   16 × 3 = 48
   48 × 7 = 336
3. Write the answer. The value of the number.
   2⁴ × 3 × 7 = 336

Check. Prime factorise 336 again and you should get 2⁴ × 3 × 7. ✓

Watch out. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.

Method. Evaluate each power first, then multiply.

1. Work out each power separately. A power means repeated multiplication of the base.
   3³ = 3 × 3 × 3 = 27
   5² = 5 × 5 = 25
2. Multiply the results together. The numbers in the product are multiplied.
   27 × 25 = 675
3. Write the answer. The value of the number.
   3³ × 5² = 675

Check. Prime factorise 675 again and you should get 3³ × 5². ✓

Watch out. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.

Method. Evaluate each power first, then multiply.

1. Work out each power separately. A power means repeated multiplication of the base.
   2³ = 2 × 2 × 2 = 8
   3 = 3
   5² = 5 × 5 = 25
2. Multiply the results together. The numbers in the product are multiplied.
   8 × 3 = 24
   24 × 25 = 600
3. Write the answer. The value of the number.
   2³ × 3 × 5² = 600

Check. Prime factorise 600 again and you should get 2³ × 3 × 5². ✓

Watch out. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. A number is a square number only if every power in its prime factorisation is even.

1. Write the number as a product of primes (index form). Use a factor tree or repeated division by primes.
   36 = 2² × 3²
2. Look at the power of each prime. A square number is a number multiplied by itself, so every prime in it comes in pairs. That means every power must be even.
   2: power 2 is even
   3: power 2 is even
3. Decide. All the powers are even, so the number is a square. Halve every power to find the square root.
   Yes: √36 = 2 × 3 = 6

Check. 6 × 6 = 36. ✓

Watch out. Look at every prime. One odd power is enough to stop it being a square.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. A number is a square number only if every power in its prime factorisation is even.

1. Write the number as a product of primes (index form). Use a factor tree or repeated division by primes.
   72 = 2³ × 3²
2. Look at the power of each prime. A square number is a number multiplied by itself, so every prime in it comes in pairs. That means every power must be even.
   2: power 3 is odd
   3: power 2 is even
3. Decide. At least one power is odd, so a prime has no partner.
   No: the power of 2 is odd.

Check. 8² = 64 and 9² = 81, so 72 is between two consecutive squares. ✓

Watch out. Look at every prime. One odd power is enough to stop it being a square.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. A number is a square number only if every power in its prime factorisation is even.

1. Write the number as a product of primes (index form). Use a factor tree or repeated division by primes.
   100 = 2² × 5²
2. Look at the power of each prime. A square number is a number multiplied by itself, so every prime in it comes in pairs. That means every power must be even.
   2: power 2 is even
   5: power 2 is even
3. Decide. All the powers are even, so the number is a square. Halve every power to find the square root.
   Yes: √100 = 2 × 5 = 10

Check. 10 × 10 = 100. ✓

Watch out. Look at every prime. One odd power is enough to stop it being a square.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. A number is a square number only if every power in its prime factorisation is even.

1. Write the number as a product of primes (index form). Use a factor tree or repeated division by primes.
   180 = 2² × 3² × 5
2. Look at the power of each prime. A square number is a number multiplied by itself, so every prime in it comes in pairs. That means every power must be even.
   2: power 2 is even
   3: power 2 is even
   5: power 1 is odd
3. Decide. At least one power is odd, so a prime has no partner.
   No: the power of 5 is odd.

Check. 13² = 169 and 14² = 196, so 180 is between two consecutive squares. ✓

Watch out. Look at every prime. One odd power is enough to stop it being a square.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. A number is a square number only if every power in its prime factorisation is even.

1. Write the number as a product of primes (index form). Use a factor tree or repeated division by primes.
   225 = 3² × 5²
2. Look at the power of each prime. A square number is a number multiplied by itself, so every prime in it comes in pairs. That means every power must be even.
   3: power 2 is even
   5: power 2 is even
3. Decide. All the powers are even, so the number is a square. Halve every power to find the square root.
   Yes: √225 = 3 × 5 = 15

Check. 15 × 15 = 225. ✓

Watch out. Look at every prime. One odd power is enough to stop it being a square.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. A number is a square number only if every power in its prime factorisation is even.

1. Write the number as a product of primes (index form). Use a factor tree or repeated division by primes.
   324 = 2² × 3⁴
2. Look at the power of each prime. A square number is a number multiplied by itself, so every prime in it comes in pairs. That means every power must be even.
   2: power 2 is even
   3: power 4 is even
3. Decide. All the powers are even, so the number is a square. Halve every power to find the square root.
   Yes: √324 = 2 × 3² = 18

Check. 18 × 18 = 324. ✓

Watch out. Look at every prime. One odd power is enough to stop it being a square.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Compare the prime factors: a number is a factor only if all its primes (with their powers) are inside the other number.

1. Write 12 as a product of primes. Compare it with the prime factors of the big number.
   12 = 2² × 3
2. Compare prime by prime. 12 divides the big number only if the big number has at least as many copies of every prime.
   2: 12 needs 2²; N has 2³  ✓
   3: 12 needs 3¹; N has 3²  ✓
3. Decide.
   Yes, 12 is a factor of N.

Check. N = 360 and 360 ÷ 12 = 30. ✓

Watch out. Compare the powers, not just which primes are there.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Compare the prime factors: a number is a factor only if all its primes (with their powers) are inside the other number.

1. Write 25 as a product of primes. Compare it with the prime factors of the big number.
   25 = 5²
2. Compare prime by prime. 25 divides the big number only if the big number has at least as many copies of every prime.
   5: 25 needs 5²; N has 5¹  ✗
3. Decide.
   No, 25 is not a factor of N.

Check. N = 360 and 360 ÷ 25 = 14 remainder 10, not exact. ✓

Watch out. Compare the powers, not just which primes are there.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Compare the prime factors: a number is a factor only if all its primes (with their powers) are inside the other number.

1. Write 40 as a product of primes. Compare it with the prime factors of the big number.
   40 = 2³ × 5
2. Compare prime by prime. 40 divides the big number only if the big number has at least as many copies of every prime.
   2: 40 needs 2³; N has 2³  ✓
   5: 40 needs 5¹; N has 5¹  ✓
3. Decide.
   Yes, 40 is a factor of N.

Check. N = 360 and 360 ÷ 40 = 9. ✓

Watch out. Compare the powers, not just which primes are there.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Compare the prime factors: a number is a factor only if all its primes (with their powers) are inside the other number.

1. Write 27 as a product of primes. Compare it with the prime factors of the big number.
   27 = 3³
2. Compare prime by prime. 27 divides the big number only if the big number has at least as many copies of every prime.
   3: 27 needs 3³; N has 3²  ✗
3. Decide.
   No, 27 is not a factor of N.

Check. N = 360 and 360 ÷ 27 = 13 remainder 9, not exact. ✓

Watch out. Compare the powers, not just which primes are there.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Compare the prime factors: a number is a factor only if all its primes (with their powers) are inside the other number.

1. Write 90 as a product of primes. Compare it with the prime factors of the big number.
   90 = 2 × 3² × 5
2. Compare prime by prime. 90 divides the big number only if the big number has at least as many copies of every prime.
   2: 90 needs 2¹; N has 2³  ✓
   3: 90 needs 3²; N has 3²  ✓
   5: 90 needs 5¹; N has 5¹  ✓
3. Decide.
   Yes, 90 is a factor of N.

Check. N = 360 and 360 ÷ 90 = 4. ✓

Watch out. Compare the powers, not just which primes are there.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Compare the prime factors: a number is a factor only if all its primes (with their powers) are inside the other number.

1. Write 16 as a product of primes. Compare it with the prime factors of the big number.
   16 = 2⁴
2. Compare prime by prime. 16 divides the big number only if the big number has at least as many copies of every prime.
   2: 16 needs 2⁴; N has 2³  ✗
3. Decide.
   No, 16 is not a factor of N.

Check. N = 360 and 360 ÷ 16 = 22 remainder 8, not exact. ✓

Watch out. Compare the powers, not just which primes are there.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 216 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

1. Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
   216 ÷ 2 = 108
   108 ÷ 2 = 54
   54 ÷ 2 = 27
   27 ÷ 3 = 9
   9 ÷ 3 = 3
   3 ÷ 3 = 1
2. Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
   216 = 2 × 2 × 2 × 3 × 3 × 3
3. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   216 = 2³ × 3³

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 3 = 24
24 × 3 = 72
72 × 3 = 216
The result is 216, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Volume of a cube = edge × edge × edge, so the edge is the cube root of the volume. With prime factors, share the primes into three equal groups.

1. Write 216 in index form. From part (a).
   216 = 2³ × 3³
2. Split into three equal groups of primes. A cube is edge × edge × edge, so each edge gets one third of the powers: one 2 and one 3.
   216 = (2 × 3) × (2 × 3) × (2 × 3)
3. Work out the edge.
   2 × 3 = 6, so the edge is 6 cm.

Check. 6 × 6 × 6 = 216. ✓

Method. The number of factors is found from the powers in the prime factorisation: add 1 to each power, then multiply.

1. Write the number in index form. The powers tell us how many times each prime can be used.
   12 = 2² × 3
2. For each prime, count the choices of how many copies to use: from none up to all of them. A power of 2² allows 2⁰, 2¹, ... 2²: that is 2 + 1 choices.
   2: 2 + 1 = 3 choices
   3: 1 + 1 = 2 choices
3. Multiply the numbers of choices. Each factor of the number is made by one choice for each prime, so we multiply (the product rule for counting).
   3 × 2 = 6

Check. List them: 1, 2, 3, 4, 6, 12 is 6 factors. ✓

Watch out. Add 1 to each power before multiplying. For 2³ there are 4 choices (2⁰, 2¹, 2², 2³), not 3.

Method. The number of factors is found from the powers in the prime factorisation: add 1 to each power, then multiply.

1. Write the number in index form. The powers tell us how many times each prime can be used.
   36 = 2² × 3²
2. For each prime, count the choices of how many copies to use: from none up to all of them. A power of 2² allows 2⁰, 2¹, ... 2²: that is 2 + 1 choices.
   2: 2 + 1 = 3 choices
   3: 2 + 1 = 3 choices
3. Multiply the numbers of choices. Each factor of the number is made by one choice for each prime, so we multiply (the product rule for counting).
   3 × 3 = 9

Check. List them: 1, 2, 3, 4, 6, 9, 12, 18, 36 is 9 factors. ✓

Watch out. Add 1 to each power before multiplying. For 2³ there are 4 choices (2⁰, 2¹, 2², 2³), not 3.

Method. The number of factors is found from the powers in the prime factorisation: add 1 to each power, then multiply.

1. Write the number in index form. The powers tell us how many times each prime can be used.
   60 = 2² × 3 × 5
2. For each prime, count the choices of how many copies to use: from none up to all of them. A power of 2² allows 2⁰, 2¹, ... 2²: that is 2 + 1 choices.
   2: 2 + 1 = 3 choices
   3: 1 + 1 = 2 choices
   5: 1 + 1 = 2 choices
3. Multiply the numbers of choices. Each factor of the number is made by one choice for each prime, so we multiply (the product rule for counting).
   3 × 2 × 2 = 12

Check. List them: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 is 12 factors. ✓

Watch out. Add 1 to each power before multiplying. For 2³ there are 4 choices (2⁰, 2¹, 2², 2³), not 3.

Method. The number of factors is found from the powers in the prime factorisation: add 1 to each power, then multiply.

1. Write the number in index form. The powers tell us how many times each prime can be used.
   100 = 2² × 5²
2. For each prime, count the choices of how many copies to use: from none up to all of them. A power of 2² allows 2⁰, 2¹, ... 2²: that is 2 + 1 choices.
   2: 2 + 1 = 3 choices
   5: 2 + 1 = 3 choices
3. Multiply the numbers of choices. Each factor of the number is made by one choice for each prime, so we multiply (the product rule for counting).
   3 × 3 = 9

Check. List them: 1, 2, 4, 5, 10, 20, 25, 50, 100 is 9 factors. ✓

Watch out. Add 1 to each power before multiplying. For 2³ there are 4 choices (2⁰, 2¹, 2², 2³), not 3.

Method. The number of factors is found from the powers in the prime factorisation: add 1 to each power, then multiply.

1. Write the number in index form. The powers tell us how many times each prime can be used.
   180 = 2² × 3² × 5
2. For each prime, count the choices of how many copies to use: from none up to all of them. A power of 2² allows 2⁰, 2¹, ... 2²: that is 2 + 1 choices.
   2: 2 + 1 = 3 choices
   3: 2 + 1 = 3 choices
   5: 1 + 1 = 2 choices
3. Multiply the numbers of choices. Each factor of the number is made by one choice for each prime, so we multiply (the product rule for counting).
   3 × 3 × 2 = 18

Check. List them: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 is 18 factors. ✓

Watch out. Add 1 to each power before multiplying. For 2³ there are 4 choices (2⁰, 2¹, 2², 2³), not 3.

Method. The number of factors is found from the powers in the prime factorisation: add 1 to each power, then multiply.

1. Write the number in index form. The powers tell us how many times each prime can be used.
   360 = 2³ × 3² × 5
2. For each prime, count the choices of how many copies to use: from none up to all of them. A power of 2³ allows 2⁰, 2¹, ... 2³: that is 3 + 1 choices.
   2: 3 + 1 = 4 choices
   3: 2 + 1 = 3 choices
   5: 1 + 1 = 2 choices
3. Multiply the numbers of choices. Each factor of the number is made by one choice for each prime, so we multiply (the product rule for counting).
   4 × 3 × 2 = 24

Check. List them: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 is 24 factors. ✓

Watch out. Add 1 to each power before multiplying. For 2³ there are 4 choices (2⁰, 2¹, 2², 2³), not 3.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Make every power in the prime factorisation even by multiplying by the primes that have an odd power.

1. Write the number in index form. We need to see the power of each prime.
   12 = 2² × 3
2. Find the primes with an odd power. A square needs every power even, so each of these needs one more copy.
   2²: even, fine
   3¹: odd, needs one more
3. Multiply the missing primes to get k. Use the smallest amount so that k is as small as possible.
   k = 3 = 3
4. Check the new number. Now all the powers are even.
   12 × 3 = 2² × 3² = 36 = 6²

Check. 6 × 6 = 36, and 36 ÷ 12 = 3. ✓

Watch out. The answer is the smallest k. Do not multiply by extra squares.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Make every power in the prime factorisation even by multiplying by the primes that have an odd power.

1. Write the number in index form. We need to see the power of each prime.
   18 = 2 × 3²
2. Find the primes with an odd power. A square needs every power even, so each of these needs one more copy.
   2¹: odd, needs one more
   3²: even, fine
3. Multiply the missing primes to get k. Use the smallest amount so that k is as small as possible.
   k = 2 = 2
4. Check the new number. Now all the powers are even.
   18 × 2 = 2² × 3² = 36 = 6²

Check. 6 × 6 = 36, and 36 ÷ 18 = 2. ✓

Watch out. The answer is the smallest k. Do not multiply by extra squares.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Make every power in the prime factorisation even by multiplying by the primes that have an odd power.

1. Write the number in index form. We need to see the power of each prime.
   50 = 2 × 5²
2. Find the primes with an odd power. A square needs every power even, so each of these needs one more copy.
   2¹: odd, needs one more
   5²: even, fine
3. Multiply the missing primes to get k. Use the smallest amount so that k is as small as possible.
   k = 2 = 2
4. Check the new number. Now all the powers are even.
   50 × 2 = 2² × 5² = 100 = 10²

Check. 10 × 10 = 100, and 100 ÷ 50 = 2. ✓

Watch out. The answer is the smallest k. Do not multiply by extra squares.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Make every power in the prime factorisation even by multiplying by the primes that have an odd power.

1. Write the number in index form. We need to see the power of each prime.
   360 = 2³ × 3² × 5
2. Find the primes with an odd power. A square needs every power even, so each of these needs one more copy.
   2³: odd, needs one more
   3²: even, fine
   5¹: odd, needs one more
3. Multiply the missing primes to get k. Use the smallest amount so that k is as small as possible.
   k = 2 × 5 = 10
4. Check the new number. Now all the powers are even.
   360 × 10 = 2⁴ × 3² × 5² = 3600 = 60²

Check. 60 × 60 = 3600, and 3600 ÷ 360 = 10. ✓

Watch out. The answer is the smallest k. Do not multiply by extra squares.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Make every power in the prime factorisation even by multiplying by the primes that have an odd power.

1. Write the number in index form. We need to see the power of each prime.
   756 = 2² × 3³ × 7
2. Find the primes with an odd power. A square needs every power even, so each of these needs one more copy.
   2²: even, fine
   3³: odd, needs one more
   7¹: odd, needs one more
3. Multiply the missing primes to get k. Use the smallest amount so that k is as small as possible.
   k = 3 × 7 = 21
4. Check the new number. Now all the powers are even.
   756 × 21 = 2² × 3⁴ × 7² = 15,876 = 126²

Check. 126 × 126 = 15,876, and 15,876 ÷ 756 = 21. ✓

Watch out. The answer is the smallest k. Do not multiply by extra squares.