GCSE Maths Factors, Multiples and Primes (Foundation): check-up and exam-style questions

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 28 (two numbers that multiply to give 28) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 28 = 28
2. Test 2, 3, 4, ... in order. Stop at 5. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 5 because 5 × 5 = 25 is the closest we can get to 28 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 28 = 28  ✓ pair: 1 and 28
   2 × 14 = 28  ✓ pair: 2 and 14
   3: 28 ÷ 3 = 9 remainder 1, so 3 is not a factor
   4 × 7 = 28  ✓ pair: 4 and 7
   5: 28 ÷ 5 = 5 remainder 3, so 5 is not a factor
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 4, 7, 14, 28

Check. 3 pairs, so 6 factors. Each factor divides 28 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.

Method. Test whether any prime up to √87 divides into 87.

1. Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.
   To show 87 is not prime we need one extra factor.
2. Decide which numbers to test. Factors come in pairs. If 87 had a factor bigger than √87, its partner would be smaller than √87 and we would meet it first. Here 9 × 9 = 81 < 87 and 10 × 10 = 100 > 87, so we only need to test the primes up to 9. We test primes only, because any other divisor is built from primes.
   Primes up to 9: 2, 3, 5, 7
3. Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.
   87 ÷ 2 = 43 remainder 1, so 2 is not a factor.
   87 ÷ 3 = 29 exactly, so 3 is a factor.
4. Decide. 3 is a factor other than 1 and 87, so there are more than two factors.
   87 = 3 × 29, so 87 is not prime.

Check. 3 × 29 = 87. ✓

Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).

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. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. The highest common factor (HCF) is the biggest number that divides into both 18 and 30. Here we use prime factors in a Venn diagram.

1. Write 18 and 30 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
   18 = 2 × 3 × 3
   30 = 2 × 3 × 5
2. Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
3. HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.
   HCF = 2 × 3 = 6

Check. HCF × LCM = 6 × 90 = 540 and 18 × 30 = 540. They are equal. ✓

Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. The lowest common multiple (LCM) is the smallest number that is in both the 8 times table and the 12 times table. Here we use prime factors in a Venn diagram.

1. Write 8 and 12 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
   8 = 2 × 2 × 2
   12 = 2 × 2 × 3
2. Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
3. LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.
   LCM = 2 × 2 × 2 × 3 = 24

Check. HCF × LCM = 4 × 24 = 96 and 8 × 12 = 96. They are equal. ✓

Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

1. Think: which number multiplied by itself gives 169? A square root undoes a square.
   We want ? × ? = 169
2. Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
   11² = 121
   12² = 144
   13² = 169
   14² = 196
   15² = 225
3. Pick the one that matches. Only one square equals the number.
   13 × 13 = 169, so √169 = 13

Check. Square the answer: 13² = 169. ✓

Watch out. √169 means the positive number 13. (The equation x² = 169 has two answers, 13 and −13, but the √ sign gives only the positive one.)

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To cube a number, multiply three copies of it together.

1. Write it as a multiplication. Cubing means multiplying three copies of the number.
   5³ = 5 × 5 × 5
2. Do the first two numbers first. Multiply in stages so that each stage is easy.
   5 × 5 = 25
3. Multiply by the third number. Now multiply the answer by the last copy.
   25 × 5 = 125

Check. Go the other way: the cube root of 125 should be 5, and 5 × 5 × 5 = 125. ✓

Watch out. 5³ means 5 × 5 × 5, not 5 × 3.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

1. Write each power out in full. 2⁶ is 6 copies of 2 multiplied, and 2³ is 3 copies.
   2⁶ = 2 × 2 × 2 × 2 × 2 × 2
   2³ = 2 × 2 × 2
2. Join them into one multiplication and count the copies of 2. Multiplying the two powers puts all the copies together: 6 + 3 copies.
   6 + 3 = 9 copies of 2
3. Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
   2⁶ × 2³ = 2⁹

Check. Work out each power: 64 × 8 = 512, which is 2⁹. ✓

Watch out. Add the indices, do not multiply them: 2⁶ × 2³ = 2⁹, not 2¹⁸. And the base stays the same.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 4.
   4³ = 64
   4² = 16
   4 = 4
   4⁰ = 1
   4⁻¹ = 14
   4⁻² = 116
2. Spot the rule. A negative index means “one over”: 4⁻² is the reciprocal of 4².
   4⁻² = 14²
3. Work out the value of the bottom. Evaluate the positive power.
   4² = 16, so 4⁻² = 116

Check. 4⁻² × 4² should be 4⁰ = 1: 116 × 16 = 1. ✓

Watch out. 4⁻² is 116, not −16. The minus sign in the index changes it to a reciprocal.

Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.

Method. Find the two whole numbers whose squares are either side of 60.

1. Find the square numbers just below and just above 60. Use the square numbers you know.
   7² = 49
   8² = 64
   49 < 60 < 64
2. Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 60 is between 7 and 8.
   7 < √60 < 8

Check. Use a calculator: √60 = 7.746..., which is between 7 and 8. ✓

Watch out. √60 is not a whole number here, so do not give an exact whole-number answer.

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 60 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.
   60 = 6 × 10
   6 = 2 × 3
   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.
   60 = 2 × 2 × 3 × 5
4. Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
   60 = 2² × 3 × 5

Check. Multiply the primes back together:
2 × 2 = 4
4 × 3 = 12
12 × 5 = 60
The result is 60, 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. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 20 (two numbers that multiply to give 20) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 20 = 20
2. Test 2, 3, 4, ... in order. Stop at 4. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 4 because 4 × 4 = 16 is the closest we can get to 20 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 20 = 20  ✓ pair: 1 and 20
   2 × 10 = 20  ✓ pair: 2 and 10
   3: 20 ÷ 3 = 6 remainder 2, so 3 is not a factor
   4 × 5 = 20  ✓ pair: 4 and 5
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 4, 5, 10, 20

Check. 3 pairs, so 6 factors. Each factor divides 20 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. The lowest common multiple (LCM) is the smallest number that is in both the 12 times table and the 15 times table. Here we list the multiples.

1. List the multiples of 12. Count up in 12s. Keep going until you have passed a number that is also in the other list.
   Multiples of 12: 12, 24, 36, 48, 60, ...
2. List the multiples of 15. Count up in 15s in the same way.
   Multiples of 15: 15, 30, 45, 60, ...
3. Find the first number that is in both lists. That is the lowest common multiple.
   LCM = 60

Check. 60 ÷ 12 = 5 and 60 ÷ 15 = 4, both whole numbers. And no smaller number appears in both lists. ✓

Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 36 and 48.

1. Write 36 and 48 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
   36 = 2 × 2 × 3 × 3
   48 = 2 × 2 × 2 × 2 × 3
2. Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
3. HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.
   HCF = 2 × 2 × 3 = 12
4. LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.
   LCM = 3 × 2 × 2 × 3 × 2 × 2 = 144

Check. HCF × LCM = 12 × 144 = 1728 and 36 × 48 = 1728. They are equal. ✓

Watch out. Check you give both answers, labelled HCF and LCM.

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

1. Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
   √(6² + 8²)
2. Work out the powers inside the root. The root sign acts like a bracket: do the working underneath it first.
   6² = 36
   8² = 64
3. Add inside the root. Add the two squares.
   36 + 64 = 100
4. Take the square root. Now find the number that squares to give the total.
   √100 = 10

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 10. ✓

Watch out. You cannot split it: √(6² + 8²) is not 6 + 8 = 14.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 2.
   2³ = 8
   2² = 4
   2 = 2
   2⁰ = 1
   2⁻¹ = 12
   2⁻² = 14
   2⁻³ = 18
2. Spot the rule. A negative index means “one over”: 2⁻³ is the reciprocal of 2³.
   2⁻³ = 12³
3. Work out the value of the bottom. Evaluate the positive power.
   2³ = 8, so 2⁻³ = 18

Check. 2⁻³ × 2³ should be 2⁰ = 1: 18 × 8 = 1. ✓

Watch out. 2⁻³ is 18, not −8. The minus sign in the index changes it to a reciprocal.

Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.

Method. Find the two whole numbers whose squares are either side of 70.

1. Find the square numbers just below and just above 70. Use the square numbers you know.
   8² = 64
   9² = 81
   64 < 70 < 81
2. Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 70 is between 8 and 9.
   8 < √70 < 9

Check. Use a calculator: √70 = 8.367..., which is between 8 and 9. ✓

Watch out. √70 is not a whole number here, so do not give an exact whole-number answer.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. The lowest common multiple (LCM) is the smallest number that is in both the 12 times table and the 15 times table. Here we use prime factors in a Venn diagram.

1. Decide what the question is asking. They ring together again after a number of minutes that is in both the 12 and 15 times tables. The first time is the lowest common multiple.
   Find the lowest common multiple of 12 and 15.
2. Write 12 and 15 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
   12 = 2 × 2 × 3
   15 = 3 × 5
3. Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
4. LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.
   LCM = 2 × 2 × 3 × 5 = 60
5. Answer the question in words. Always finish by saying what the number means.
   They ring together again after 60 minutes, so at 11:00 am.

Check. HCF × LCM = 3 × 60 = 180 and 12 × 15 = 180. They are equal. ✓

Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. The highest common factor (HCF) is the biggest number that divides into both 168 and 252. Here we use prime factors in a Venn diagram.

1. Write 168 and 252 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
   168 = 2 × 2 × 2 × 3 × 7
   252 = 2 × 2 × 3 × 3 × 7
2. Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
3. HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.
   HCF = 2 × 2 × 3 × 7 = 84
4. Answer the question in words. Always finish by saying what the number means.
   The HCF of 168 and 252 is 84.

Check. HCF × LCM = 84 × 504 = 42336 and 168 × 252 = 42336. They are equal. ✓

Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.