Year 5: Putting it all together: clues and reasoning
Part of the Year 5 prime numbers, factors, squares and cubes chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Putting it all together: clues and reasoning
In this section you use everything from the chapter. Many problems give clues about a mystery number. Here is a good way to solve them.
- Read all the clues. Choose the clue that gives the shortest list, for example “a square number” or “a prime number”.
- Write the list of numbers that fit this clue (inside the range).
- Use the other clues, one at a time, to cross out numbers.
- Check the last number left against every clue.
I am a square numberI am a multiple of 3I am between 30 and 70
The squares between 30 and 70 are 36, 49 and 64. Only 36 is a multiple of 3. So the number is 36.
How many factors? Here is a hundred-square style table of the numbers from 1 to 50. Numbers with exactly 2 factors (the primes) are gold. Numbers with exactly 3 factors are green. Numbers with exactly 4 factors are blue. Can you see a pattern? The green numbers are 4, 9, 25 and 49. They are the squares of prime numbers (2², 3², 5², 7²). The number with the most factors under 50 is 48. It has 10: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
Reasoning. The big reasoning questions are about what is always, sometimes or never true. To show “sometimes” you need one example that works and one that does not. An example that breaks a rule is a counter-example. To prove something is always true, explain why in words: examples alone are not enough. To spot a mistake, work the question out yourself first, then compare.
Key ideas
- For a clue puzzle, start with the clue that gives the shortest list. Cross out numbers with each other clue. Check the answer against every clue.
- Numbers with exactly 2 factors are prime. Numbers with exactly 3 factors are the squares of primes (4, 9, 25, 49).
- To show “sometimes” give one example that works and one that does not. One counter-example shows “always” is false.
- A proof needs a reason that works for every number, not just a few examples.
- For a problem about things that repeat (buses, lights, bells), use common multiples. For sharing into equal groups, use common factors.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a square number”. List the numbers between 30 and 70 that fit this clue.36, 49, 64
- Use the next clue: “a multiple of 3”. Cross out every number that does not fit.36,
49,64 - Check the answer against every clue. It must fit all of them.“a square number” ✓
“a multiple of 3” ✓
Check. Only 36 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a prime number”. List the numbers between 20 and 30 that fit this clue.23, 29
- Use the next clue: “my digits add up to 11”. Cross out every number that does not fit.
23, 29 - Check the answer against every clue. It must fit all of them.“a prime number” ✓
“my digits add up to 11” ✓
Check. Only 29 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a multiple of 9”. List the numbers less than 40 that fit this clue.9, 18, 27, 36
- Use the next clue: “a multiple of 4”. Cross out every number that does not fit.
9,18,27, 36 - Use the next clue: “a multiple of 6”. Cross out every number that does not fit.36
- Check the answer against every clue. It must fit all of them.“a multiple of 9” ✓
“a multiple of 4” ✓
“a multiple of 6” ✓
Check. Only 36 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
Method. Count the factors of each number. Write down the ones with exactly 3 factors.
- Find how many factors each number has. Use factor pairs. The list shows number:count.2:2 3:2 4:3 5:2 6:4 7:2 8:4 9:3 10:4 11:2
12:6 13:2 14:4 15:4 16:5 17:2 18:6 19:2 20:6 21:4
22:4 23:2 24:8 25:3 26:4 27:4 28:6 29:2 30:8 31:2
32:6 33:4 34:4 35:4 36:9 37:2 38:4 39:4 40:8 41:2
42:8 43:2 44:6 45:6 46:4 47:2 48:10 49:3 - Pick the numbers with exactly 3 factors. They are in bold.4, 9, 25, 49
- Look for a pattern. Each of these is the square of a prime number (2², 3², 5², …).
Check. Check one: 49 has factors 1, 7, 49, which is 3. ✓
Watch out. Count 1 and the number itself too.
Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
Method. Count the factors of every number in the range. To save time, notice that numbers with lots of factors are multiples of small numbers like 6 or 12. Make a table of the ones with the most.
- Count the factors of each number. Use factor pairs. Most numbers have only a few factors.Every number from 1 to 49 that is not in the table below has 6 factors or fewer.
- List the numbers with 8 or more factors. These are the only ones that could win.
Number Factors How many 24 1, 2, 3, 4, 6, 8, 12, 24 8 30 1, 2, 3, 5, 6, 10, 15, 30 8 36 1, 2, 3, 4, 6, 9, 12, 18, 36 9 40 1, 2, 4, 5, 8, 10, 20, 40 8 42 1, 2, 3, 6, 7, 14, 21, 42 8 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 10 - Find the most. Look at the last column.The most is 10 factors: 48
Check. Check one: 48 = 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8, which gives 10 factors. ✓
Watch out. Count 1 and the number itself. Do not stop at the first few pairs.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of the biggest number. Test each one against the other numbers. The first one that is in every times table is the LCM.
- List the multiples of 10. The common multiple must be in the times table of the biggest number, so we only test these.10, 20, 30, 40, 50, 60
- Test each one. A number passes if it divides exactly by 4 and by 6.10: not in the 4 times table (10 ÷ 4 = 2 remainder 2)
20: not in the 6 times table (20 ÷ 6 = 3 remainder 2)
30: not in the 4 times table (30 ÷ 4 = 7 remainder 2)
40: not in the 6 times table (40 ÷ 6 = 6 remainder 4)
50: not in the 4 times table (50 ÷ 4 = 12 remainder 2)
60: in all the tables - Write the answer. We stop at the first number that passes every test.The LCM of 4, 6 and 10 is 60.
Check. 60 ÷ 4 = 15 , 60 ÷ 6 = 10 , 60 ÷ 10 = 6. All exact. ✓
Watch out. Test every number, not just one of them.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.6 = 3 × 2, so 6 is in the 3 times table. Multiples of 6: 6, 12, 18, 24, 30. Each is also in the 3 times table: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.6 = 3 × 2. A multiple of 6 is 6 × (any number) = 3 × 2 × (any number) = 3 × (something). So it is always in the 3 times table.
- Decide. Always
Check. 42 = 6 × 7 = 3 × 14. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.Multiples of 4: 4, 8, 12, 16, 20
Multiples of 10: 10, 20
The first number in both lists is 20. - Compare with the answer in the question. Jack said 40.
- Find the mistake. 40 is a common multiple (4 × 10 = 40), but it is not the lowest. 20 is in both lists and it is smaller.
- Write the correct answer. No. The LCM of 4 and 10 is 20.
Check. 20 ÷ 4 = 5 and 20 ÷ 10 = 2. ✓
Watch out. Multiplying the two numbers does not always give the lowest common multiple.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Look for something three numbers share. Try square numbers.
- Write each as a multiplication. See whether it is a number multiplied by itself.16 = 4 × 4
25 = 5 × 5
36 = 6 × 6
48 = 6 × 8 (not equal numbers) - Decide. 16, 25 and 36 are square numbers. 48 is not.48 is the odd one out.
- Check the squares. Make sure 48 is not a square: 6² = 36 and 7² = 49.48 is between 36 and 49, so it is not a square.
Check. The squares are 1, 4, 9, 16, 25, 36, 49, … and 48 is not in the list. ✓
Watch out. There may be another reason too (25 is the only odd number here). Say which reason you use. Both can be right.
Watch out
Do not stop when you find a number that fits the first clue. Check it against all the clues. And in an “always, sometimes, never” question, one example is never enough to prove “always”. Remember “between 30 and 70” does not include 30 or 70.Exercise 3I Number clues and reasoning
Non-calculatorKey words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a multiple of 7”. List the numbers between 50 and 80 that fit this clue.56, 63, 70, 77
- Use the next clue: “an odd number”. Cross out every number that does not fit.
56, 63,70, 77 - Use the next clue: “my digits add up to 9”. Cross out every number that does not fit.63,
77 - Check the answer against every clue. It must fit all of them.“a multiple of 7” ✓
“an odd number” ✓
“my digits add up to 9” ✓
Check. Only 63 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a factor of 36”. List the numbers between 8 and 15 that fit this clue.9, 12
- Use the next clue: “an even number”. Cross out every number that does not fit.
9, 12 - Use the next clue: “a multiple of 3”. Cross out every number that does not fit.12
- Check the answer against every clue. It must fit all of them.“a factor of 36” ✓
“an even number” ✓
“a multiple of 3” ✓
Check. Only 12 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a prime number”. List the numbers between 50 and 70 that fit this clue.53, 59, 61, 67
- Use the next clue: “my digits add up to 8”. Cross out every number that does not fit.53,
59,61,67 - Check the answer against every clue. It must fit all of them.“a prime number” ✓
“my digits add up to 8” ✓
Check. Only 53 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a square number”. List the numbers between 100 and 1,000 that fit this clue.121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961
- Use the next clue: “a multiple of 5”. Cross out every number that does not fit.
121,144,169,196, 225,256,289,324,361, 400,441,484,529,576, 625,676,729,784,841, 900,961 - Use the next clue: “a multiple of 6”. Cross out every number that does not fit.
225,400,625, 900 - Check the answer against every clue. It must fit all of them.“a square number” ✓
“a multiple of 5” ✓
“a multiple of 6” ✓
Check. Only 900 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
Method. Count the factors of each number. Write down the ones with exactly 2 factors.
- Find how many factors each number has. Use factor pairs. The list shows number:count.2:2 3:2 4:3 5:2 6:4 7:2 8:4 9:3 10:4 11:2
12:6 13:2 14:4 15:4 16:5 17:2 18:6 19:2 20:6 - Pick the numbers with exactly 2 factors. They are in bold.2, 3, 5, 7, 11, 13, 17, 19
- Look for a pattern. These numbers each have exactly two factors, so they are the prime numbers.
Check. Check one: 19 has factors 1, 19, which is 2. ✓
Watch out. Count 1 and the number itself too.
Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
Method. Count the factors of each number. Write down the ones with exactly 3 factors.
- Find how many factors each number has. Use factor pairs. The list shows number:count.2:2 3:2 4:3 5:2 6:4 7:2 8:4 9:3 10:4 11:2
12:6 13:2 14:4 15:4 16:5 17:2 18:6 19:2 20:6 21:4
22:4 23:2 24:8 25:3 26:4 27:4 28:6 29:2 30:8 - Pick the numbers with exactly 3 factors. They are in bold.4, 9, 25
- Look for a pattern. Each of these is the square of a prime number (2², 3², 5², …).
Check. Check one: 25 has factors 1, 5, 25, which is 3. ✓
Watch out. Count 1 and the number itself too.
Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
Method. Count the factors of each number. Write down the ones with exactly 4 factors.
- Find how many factors each number has. Use factor pairs. The list shows number:count.2:2 3:2 4:3 5:2 6:4 7:2 8:4 9:3 10:4 11:2
12:6 13:2 14:4 15:4 16:5 17:2 18:6 19:2 20:6 - Pick the numbers with exactly 4 factors. They are in bold.6, 8, 10, 14, 15
- Look for a pattern. Each of these is a product of two different primes (like 2 × 3), or a prime cubed (like 2³).
Check. Check one: 15 has factors 1, 3, 5, 15, which is 4. ✓
Watch out. Count 1 and the number itself too.
Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
Method. Count the factors of every number in the range. To save time, notice that numbers with lots of factors are multiples of small numbers like 6 or 12. Make a table of the ones with the most.
- Count the factors of each number. Use factor pairs. Most numbers have only a few factors.Every number from 51 to 99 that is not in the table below has 8 factors or fewer.
- List the numbers with 10 or more factors. These are the only ones that could win.
Number Factors How many 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 12 72 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 12 80 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 10 84 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 12 90 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 12 96 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 12 - Find the most. Look at the last column.The most is 12 factors: 60, 72, 84, 90, 96
Check. Check one: 60 = 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10, which gives 12 factors. ✓
Watch out. Count 1 and the number itself. Do not stop at the first few pairs.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.
- Write down when each one happens. They start together at 0. The first lighthouse flashs every 6 seconds and the second lighthouse every 9 seconds.Every 6: 6, 12, 18
Every 9: 9, 18 - Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.Together at: 18
- Pick the first one. The first time they are together again is the lowest common multiple.LCM of 6 and 9 = 18
- Write the answer. 18 seconds after the start
Check. 18 ÷ 6 = 3 and 18 ÷ 9 = 2, so both happen exactly at 18. ✓
Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.
- Write down when each one happens. They start together at 0. The first lighthouse flashs every 6 seconds and the second lighthouse every 9 seconds.Every 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120
Every 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117 - Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.Together at: 18, 36, 54, 72, 90, 108
- Count them. Count the times in both lists. Do not count the start (0).18, 36, 54, 72, 90, 108 → 6 times
Check. 18 ÷ 6 = 3 and 18 ÷ 9 = 2, so both happen exactly at 18. ✓
Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Write the multiples of the biggest number. Test each one against the other numbers. The first one that is in every times table is the LCM.
- List the multiples of 6. The common multiple must be in the times table of the biggest number, so we only test these.6, 12, 18, 24, 30
- Test each one. A number passes if it divides exactly by 3 and by 5.6: not in the 5 times table (6 ÷ 5 = 1 remainder 1)
12: not in the 5 times table (12 ÷ 5 = 2 remainder 2)
18: not in the 5 times table (18 ÷ 5 = 3 remainder 3)
24: not in the 5 times table (24 ÷ 5 = 4 remainder 4)
30: in all the tables - Write the answer. We stop at the first number that passes every test.The LCM of 3, 5 and 6 is 30.
Check. 30 ÷ 3 = 10 , 30 ÷ 5 = 6 , 30 ÷ 6 = 5. All exact. ✓
Watch out. Test every number, not just one of them.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 93
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 93 is 3. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 9 + 3 = 12. 12 is in the 3 times table (3 × 4 = 12). yes. - Decide. 93 = 3 × 31, so 93 has other factors. 93 is not prime.
Check. 3 × 31 = 93. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.8 is a multiple of 4 and of 8. ✓ 16 is a multiple of 4 and of 8. ✓
- Look for one that does not work. One example that breaks the rule is called a counter-example.12 is a multiple of 4 (4 × 3) but 12 ÷ 8 = 1 remainder 4, so it is not a multiple of 8.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.The multiples of 8 are every second multiple of 4 (8, 16, 24…). The others (4, 12, 20…) are multiples of 4 only.
- Decide. Sometimes
Check. 8 is in both lists and 12 is only in the 4s. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Every factor of 12 is in the list for 24.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.24 = 12 × 2. If a number divides into 12, it divides into 12 × 2 = 24 as well. So every factor of 12 is a factor of 24.
- Decide. Always
Check. Test one: 6 divides 12 (12 ÷ 6 = 2) and 24 (24 ÷ 6 = 4). ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.1³ = 1 and 3³ = 27 are odd cube numbers.
- Look for one that does not work. One example that breaks the rule is called a counter-example.2³ = 8 and 4³ = 64 are even cube numbers.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.A cube of an odd number is odd (odd × odd × odd is odd). A cube of an even number is even.
- Decide. Sometimes
Check. 27 is odd and 64 is even. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.
Method. Look for something that three of the numbers have in common. Try square numbers.
- Write each as a multiplication. See if it is a number multiplied by itself.16 = 4 × 4
36 = 6 × 6
64 = 8 × 8
50 = 5 × 10 (not equal numbers) - Decide. Three numbers are square numbers.16, 36 and 64 are squares. 50 is not: 7² = 49 and 8² = 64.
Check. 50 is between 49 = 7² and 64 = 8². ✓
Watch out. All four numbers are even, so “even” is not the reason.
Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.Multiples of 5: 5, 10, 15, 20, 25, 30
Multiples of 6: 6, 12, 18, 24, 30
The first number in both lists is 30. - Compare with the answer in the question. Zainab said 11.
- Find the mistake. Zainab added the numbers. A common multiple is in both times tables. 11 is not in the 5 times table or in the 6 times table.
- Write the correct answer. No. The LCM of 5 and 6 is 30.
Check. 30 ÷ 5 = 6 and 30 ÷ 6 = 5. ✓
Watch out. The LCM is a multiple, so it is at least as big as the biggest number. Adding does not make a multiple.
Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
Method. Count the factors of each number. Write down the ones with exactly 5 factors.
- Find how many factors each number has. Use factor pairs. The list shows number:count.2:2 3:2 4:3 5:2 6:4 7:2 8:4 9:3 10:4 11:2
12:6 13:2 14:4 15:4 16:5 17:2 18:6 19:2 20:6 21:4
22:4 23:2 24:8 25:3 26:4 27:4 28:6 29:2 30:8 31:2
32:6 33:4 34:4 35:4 36:9 37:2 38:4 39:4 40:8 41:2
42:8 43:2 44:6 45:6 46:4 47:2 48:10 49:3 - Pick the numbers with exactly 5 factors. They are in bold.16
- Look for a pattern. Numbers with five factors are rare.
Check. Check one: 16 has factors 1, 2, 4, 8, 16, which is 5. ✓
Watch out. Count 1 and the number itself too.