Multiples, divisibility, common multiples, and LCM by listing · Grade 1–4.
This booklet helps you learn about multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12., step by step.
What’s a multiple?
A multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. is what you get when you times a number by another whole numberA number with no fractional part: 0, 1, 2, 3, 4, …. You can also think of it as the numbers you reach by counting in steps.
Example: the multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 5 are 5, 10, 15, 20, 25, 30, … — every number in the 5 times tableThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, …. They’re the numbers you reach by counting in fives.
What you’ll learn
We start with listing the multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of small numbers. Then we look at common multiplesA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. — numbers that appear in two different times tablesThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, … at the same time.
How to use the booklet
Work through the questions in order.
If you get stuck, that’s OK — stop there. Where you stop tells your tutor exactly where you need help.
Lesson 1
What a multiple is
Aim: List the multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of any whole numberA number with no fractional part: 0, 1, 2, 3, 4, …, and tell multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. apart from factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
The multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of n are: n, 2 × n, 3 × n, 4 × n, … — the same list as the n times tableThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, ….
Key fact 1: A number is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of itself. The first multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 5 is 5 (because 1 × 5 = 5). The first multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 11 is 11. And so on.
Key fact 2: Every number has infinitely many multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12.. The multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3 are 3, 6, 9, 12, 15, …, 300, 303, … They never end.
Worked example
List the first five multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 4.
1 × 4 = 4
2 × 4 = 8
3 × 4 = 12
4 × 4 = 16
5 × 4 = 20
So the first five multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 4 are 4, 8, 12, 16, 20.
Worked example
Is 30 a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6?
Divide: 30 ÷ 6 = 5 (a whole numberA number with no fractional part: 0, 1, 2, 3, 4, …, no remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1.).
So yes, 30 is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6 (because 5 × 6 = 30).
Another way: count up in 6s — 6, 12, 18, 24, 30. We reach 30, so it’s a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12..
Watch out — multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. vs factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.
This is the most common confusion in Year 7. They sound similar but mean opposite things.
A factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 12 is a number that goes INTO 12: 1, 2, 3, 4, 6, 12.
A multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 12 is a number that 12 goes INTO: 12, 24, 36, 48, …
Memory trick:factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. are smaller (or equal to) the number; multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. are bigger (or equal).
Try one
List the first five multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 7.
Show answer
Multiply 7 by 1, 2, 3, 4, 5.
7, 14, 21, 28, 35.
Check: each is 7 more than the one before. ✓
Lesson 2
Times tables — the backbone
Aim: Recall multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of any single-digitOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7. number without working them out.
To work with multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. quickly, you need the times tablesThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, … 2 through 12 in your head. Here is the full grid:
×
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10
12
14
16
18
20
22
24
3
3
6
9
12
15
18
21
24
27
30
33
36
4
4
8
12
16
20
24
28
32
36
40
44
48
5
5
10
15
20
25
30
35
40
45
50
55
60
6
6
12
18
24
30
36
42
48
54
60
66
72
7
7
14
21
28
35
42
49
56
63
70
77
84
8
8
16
24
32
40
48
56
64
72
80
88
96
9
9
18
27
36
45
54
63
72
81
90
99
108
10
10
20
30
40
50
60
70
80
90
100
110
120
11
11
22
33
44
55
66
77
88
99
110
121
132
12
12
24
36
48
60
72
84
96
108
120
132
144
square numbers (n × n)the hardest facts
The grid is symmetrical across the diagonal: 3 × 7 = 7 × 3. So you really only need to learn one triangle of facts.
Patterns to help you learn:
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 2: end in 0, 2, 4, 6, or 8 (the evenA whole number divisible by 2. The even numbers are 2, 4, 6, 8, …digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7.)
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 5: end in 0 or 5
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 10: end in 0
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 9 (up to 90): the digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7. add to 9. 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 11 (up to 99): both digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7. are the same. 11, 22, 33, 44, 55, 66, 77, 88, 99
Worked example
What is the 7th multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 8?
7 × 8 = 56.
So the 7th multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 8 is 56. (This is the hardest single times-table fact — worth knowing by sight.)
Watch out — the 7 and 12 tables
The 7 times tableThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, … and the 12 times tableThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, … are the ones pupils get wrong most often. Practise these until 7×6, 7×8, 12×7, and 12×8 are automatic.
Try one
What is 6 × 7?
Show answer
6 × 7 = 42.
Check: it equals 7 × 6 — the grid is symmetrical. ✓
Lesson 3
Common multiples
Aim: Find a number that is in two times tablesThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, … at once.
A common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. of two numbers is a number that appears in BOTH of their multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. lists.
Example: MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, … MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6: 6, 12, 18, 24, 30, 36, …
Numbers in BOTH lists: 12, 24, 36, … — these are the common multiplesA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. of 4 and 6.
The method:
List the first several multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of each number.
Find numbers that appear in both lists.
Worked example
Find a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. of 3 and 5.
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 5: 5, 10, 15, 20, 25, 30, …
Both lists contain 15 and 30 (and 45, 60, …).
So 15 is a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. of 3 and 5. (And so are 30, 45, 60, …)
Quick shortcut (use with care): For any two numbers a and b, the productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. a × b is always a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4.. So 3 × 5 = 15 is a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. of 3 and 5. But this is NOT always the smallest one. For 4 and 6, the productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. is 24 — a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4., but 12 is smaller.
Watch out
“Common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4.” just means “a number that is in both lists” — it does NOT mean “the smallest such number”. The smallest one has a special name (LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.), covered in Lesson 4.
Try one
Find a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. of 4 and 6.
Show answer
A common multiple appears in both times tables.
The smallest is 12 (also 24, 36, 48 …).
Any number in both lists is correct. ✓
Lesson 4
Lowest common multiple (LCM)
Aim: Find the smallest number that is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of two given numbers.
In Lesson 3 you found common multiplesA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. of two numbers — the numbers that appear in both times tablesThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, …. Usually there are lots of them. The lowest common multipleThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables. (LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.) is just the smallest one.
For 4 and 6, the common multiplesA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. are 12, 24, 36, … The smallest is 12, so LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.(4, 6) = 12.
Key fact 3: When two numbers share no factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. (other than 1), their LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables. equals their productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4..
Example: LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.(5, 7) = 5 × 7 = 35 (because 5 and 7 share no factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.).
Key fact 4: When two numbers share a factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., their LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables. is smaller than their productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4..
Example: LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.(4, 6) = 12 (not 24), because 4 and 6 share the factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. 2.
The reliable method (lists):
List multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of each number until you find one that appears in both lists.
The first match is the LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables..
Worked example
Find LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.(8, 12).
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 8: 8, 16, 24, 32, 40, …
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 12: 12, 24, 36, …
First number in both lists is 24. So LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.(8, 12) = 24.
Notice: 8 × 12 = 96, which is a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. but NOT the smallest. The shortcut overshoots because 8 and 12 share the factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. 4.
Worked example
Find LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.(3, 8).
3 and 8 share no factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. (3 = primeA whole number bigger than 1 with exactly 2 factors: 1 and itself., 8 = 2×2×2).
So LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables. = 3 × 8 = 24.
Check: 24 is in 3’s table (3×8=24) and in 8’s table (8×3=24). ✓
Watch out
Do not apply the multiply-together shortcut blindly. If the numbers share a factorA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. (like 4 and 6 share 2), the shortcut gives a common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. but NOT the smallest one. When in doubt, use the lists method.
Try one
Find LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables.(6, 9).
Show answer
Multiples of 6: 6, 12, 18; of 9: 9, 18, 27.
First shared: LCM = 18.
6×9 = 54 overshoots because 6 and 9 share a factor. ✓
Lesson 5
Divisibility tests — spotting multiples by sight
Aim: Decide whether a number is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 2, 3, 4, 5, 6, 9, or 10 without doing any division.
So far you’ve been listing multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. or dividing to check. That works, but it’s slow for big numbers — imagine checking if 4,572 is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3 by listing! There’s a faster way for some common numbers: you can spot multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. just by looking at the digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7..
The tests:
÷2 — ends in 0, 2, 4, 6, or 8.
÷5 — ends in 0 or 5.
÷10 — ends in 0.
÷3 — add the digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7.. If the total is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3, the original is too.
÷9 — add the digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7.. If the total is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 9, the original is too.
÷6 — must pass BOTH the ÷2 test AND the ÷3 test.
÷4 — look at the last two digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7.. If they form a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 4, the whole numberA number with no fractional part: 0, 1, 2, 3, 4, … is.
Worked example
Is 540 a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 9?
Add the digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7.: 5 + 4 + 0 = 9.
9 is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 9. So yes, 540 is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 9. (Check: 540 ÷ 9 = 60.)
Worked example
Is 132 a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6?
Check ÷2: last digitOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7. is 2, so 132 is evenA whole number divisible by 2. The even numbers are 2, 4, 6, 8, …. ✓
Check ÷3: digit sumThe total when you add up the digits. The digit sum of 234 is 2 + 3 + 4 = 9. 1 + 3 + 2 = 6, a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3. ✓
Both tests pass, so 132 is a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6. (Check: 132 ÷ 6 = 22.)
Watch out — don’t mix the rules
The digit-sumThe total when you add up the digits. The digit sum of 234 is 2 + 3 + 4 = 9. rule is only for 3 and 9. Don’t use it for 4, 7, or 8. For 4, use the last-two-digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7. rule.
Try one
Is 825 a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 5? Is 825 a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3?
Show answer
Ends in 5, so it is a multiple of 5.
Digit sum 8+2+5 = 15, a multiple of 3, so it is a multiple of 3.
So yes (825 = 3 × 275 = 5 × 165). ✓
VOCABULARY
Multiple
A number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12.
Factor
A number that divides another exactly. 3 is a factor of 12 because 12 ÷ 3 = 4 (whole).
Times table
The list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, …
Common multiple
A number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4.
Lowest common multiple (LCM)
The smallest common multiple of two numbers. LCM(4, 6) = 12.
Digit sum
The total when you add up the digits. The digit sum of 234 is 2 + 3 + 4 = 9.
Divisible by
“36 is divisible by 4” means 36 ÷ 4 gives a whole number (no remainder).
Recap before you start the paper
MultipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of n = a number in the n times tableThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, … (n, 2n, 3n, …).
Every number is its own first multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12..
MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. vs factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.:factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. go INTO n; multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. are what n goes INTO.
Common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4. = a number in BOTH times tablesThe list of multiples of a number. The 3 times table is 3, 6, 9, 12, 15, ….
LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables. = the smallest common multipleA number that appears in two (or more) times tables. 12 is a common multiple of 3 and 4.. For numbers that share no factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., LCMThe smallest common multiple of two numbers. LCM(4, 6) = 12, the smallest number in both the 4 and 6 times tables. = their productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4..
Divisibility tests: ÷2 (ends 0,2,4,6,8), ÷5 (ends 0,5), ÷10 (ends 0), ÷3 / ÷9 (digit sumThe total when you add up the digits. The digit sum of 234 is 2 + 3 + 4 = 9.), ÷6 (both ÷2 AND ÷3), ÷4 (last two digitsOne of the symbols 0–9 used to write numbers. The number 87 has two digits: 8 and 7.).
Common traps: bigger doesn’t mean a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12.. Round doesn’t mean a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12.. The multiply-together shortcut overshoots when the numbers share factorsA number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
Part Two
Now try the practice questions
45 questions across four sections. No time pressure. The step-by-step walkthroughs unlock after you mark.
Instructions
Type each numeric answer in the box. For sign questions, use a minus sign (e.g. -7). For radio questions, tap your choice.
Work through the questions carefully. There is no time pressure.
Your tutor is looking for understanding, method, and the point where you begin to struggle.
When finished, click Mark My Work. Your answers will then be locked, the correct answers will appear under each wrong question with the working, and you will get 5 similar practice questions inline for each one you missed.
Take your time
No time pressure
Work carefully. Your tutor is looking for understanding, method, and the point where you begin to struggle.
Section A · Recognising and listing multiples
Recognise multiples, list them, count them in a range, and use divisibility tests.
LEARNING OBJECTIVE 1 · GRADE 1
Recognise multiples of a number and list them systematically.
Success criteria — I can:
list the first few multiples of a number (e.g. multiples of 7: 7, 14, 21, 28, …)
decide whether a given number is a multiple of another
use divisibility tests for 2, 3, 5, and 10
count how many multiples of n fall in a given range
Q1.
Is 24 a multiple of 6?
Your answer:(1 mark)
We need to check if 24 is a multiple of 6.
A multiple of 6 means a number you get by multiplying 6 by a whole number.
Try dividing: 24 ÷ 6 = 4.
4 is a whole number with no remainder.
So 24 IS a multiple of 6, because 4 × 6 = 24.
Another way to check: count up in 6s — 6, 12, 18, 24. We reach 24. ✓
Q2.
Is 45 a multiple of 9?
Your answer:(1 mark)
We need to check if 45 is a multiple of 9.
Try dividing: 45 ÷ 9 = 5.
5 is a whole number with no remainder.
So 45 IS a multiple of 9, because 5 × 9 = 45.
Quick check using the digit-sum rule for ÷9: 4 + 5 = 9.
9 is a multiple of 9, so 45 must be too. ✓
Q3.
Is 38 a multiple of 4?
Your answer:(1 mark)
We need to check if 38 is a multiple of 4.
Try dividing: 38 ÷ 4 = 9.5.
9.5 is NOT a whole number (it has a decimal).
So 38 is NOT a multiple of 4.
Check using the last-two-digits rule for ÷4: the last two digits of 38 are 38 itself.
Is 38 in the 4 times table? 4 × 9 = 36, 4 × 10 = 40. 38 is between two multiples of 4, but isn’t one. ✗
Watch out: 38 is even, but even numbers aren’t always multiples of 4. Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
Q4.
Is 100 a multiple of 7?
Your answer:(1 mark)
We need to check if 100 is a multiple of 7.
Try dividing: 100 ÷ 7 = 14.28...
14.28... is NOT a whole number.
So 100 is NOT a multiple of 7.
Find the closest multiples of 7: 7 × 14 = 98, and 7 × 15 = 105.
100 sits between 98 and 105 — it’s not in the 7 times table. ✗
Watch out: 100 looks like a special round number, but that does not make it a multiple of 7.
Q5.
Is 144 a multiple of 12?
Your answer:(1 mark)
We need to check if 144 is a multiple of 12.
Try dividing: 144 ÷ 12 = 12.
12 is a whole number with no remainder.
So 144 IS a multiple of 12, because 12 × 12 = 144.
144 is the 12th multiple of 12. (It is also a square number, because 12 × 12 = 144.) ✓
Q6.
What is the 3rd multiple of 5?
Your answer:(1 mark)
The Nth multiple of a number means N times that number.
The 3rd multiple of 5 means 3 × 5.
3 × 5 = 15.
Check by counting: 1st multiple is 5, 2nd is 10, 3rd is 15. ✓
Q7.
What is the 7th multiple of 8?
Your answer:(1 mark)
The 7th multiple of 8 means 7 × 8.
7 × 8 = 56.
Check by counting: 8, 16, 24, 32, 40, 48, 56. ✓
(This is the hardest single times-table fact in Year 7 — worth memorising.)
Q8.
What is the 10th multiple of 4?
Your answer:(1 mark)
The 10th multiple of 4 means 10 × 4.
10 × 4 = 40.
Multiplying by 10 just adds a zero to the end — so this is an easy one to spot. ✓
Q9.
How many multiples of 6 are there from 1 to 30 (including 30)?
Your answer:(2 marks)
We need to count how many multiples of 6 sit from 1 to 30.
List them: 6, 12, 18, 24, 30.
Count: 5 multiples.
Shortcut: 30 ÷ 6 = 5. Because 30 is the 5th multiple of 6, there are 5 multiples of 6 up to and including 30. ✓
Q10.
How many multiples of 9 are there from 1 to 100?
Your answer:(2 marks)
We need to count how many multiples of 9 sit from 1 to 100.
Shortcut: 100 ÷ 9 = 11.11..., and we take only the whole-number part — 11.
Watch out: a common mistake is to answer 10. The 11th multiple is 99, which IS below 100, so 11 is correct. ✓
Q11.
Is 234 a multiple of 3?
Your answer:(1 mark)
We need to check if 234 is a multiple of 3.
Use the digit-sum rule for ÷3 (from Lesson 5).
Add the digits: 2 + 3 + 4 = 9.
Is 9 a multiple of 3? Yes (3 × 3 = 9).
So 234 IS a multiple of 3.
Check by dividing: 234 ÷ 3 = 78 (a whole number). ✓
Q12.
Is 248 a multiple of 4?
Your answer:(2 marks)
We need to check if 248 is a multiple of 4.
Use the last-two-digits rule for ÷4.
The last two digits of 248 are 48.
Is 48 a multiple of 4? Yes (4 × 12 = 48).
So 248 IS a multiple of 4.
Check by dividing: 248 ÷ 4 = 62. ✓
Q13.
Is 195 a multiple of 6?
Your answer:(2 marks)
We need to check if 195 is a multiple of 6.
To be a multiple of 6, a number must pass BOTH the ÷2 test AND the ÷3 test.
Check ÷2: does 195 end in 0, 2, 4, 6, or 8? The last digit is 5. NO — 195 fails the ÷2 test.
Even though the digit sum (1 + 9 + 5 = 15) is a multiple of 3, the ÷2 test failed.
So 195 is NOT a multiple of 6. ✗
Watch out: you must check BOTH rules for ÷6. Failing either one rules it out.
Q14.
Is 567 a multiple of 9?
Your answer:(2 marks)
We need to check if 567 is a multiple of 9.
Use the digit-sum rule for ÷9.
Add the digits: 5 + 6 + 7 = 18.
Is 18 a multiple of 9? Yes (9 × 2 = 18).
So 567 IS a multiple of 9.
Check by dividing: 567 ÷ 9 = 63. ✓
Q15.
Is 220 a multiple of 4?
Your answer:(1 mark)
We need to check if 220 is a multiple of 4.
Use the last-two-digits rule for ÷4.
The last two digits of 220 are 20.
Is 20 a multiple of 4? Yes (4 × 5 = 20).
So 220 IS a multiple of 4.
Check: 220 = 4 × 55. ✓
Q16.
List the first four multiples of 7.
Your answer:
,,,
(4 marks)
The multiples of 7 come from 1×7, 2×7, 3×7, …
1st multiple: 1 × 7 = 7.
2nd multiple: 2 × 7 = 14.
3rd multiple: 3 × 7 = 21.
4th multiple: 4 × 7 = 28.
So the first four multiples of 7 are: 7, 14, 21, 28. ✓
Q17.
List the first three multiples of 12.
Your answer:
,,
(3 marks)
1st multiple: 1 × 12 = 12.
2nd multiple: 2 × 12 = 24.
3rd multiple: 3 × 12 = 36.
So the first three multiples of 12 are: 12, 24, 36. ✓
Q18.
List the first three multiples of 11.
Your answer:
,,
(3 marks)
1st multiple: 1 × 11 = 11.
2nd multiple: 2 × 11 = 22.
3rd multiple: 3 × 11 = 33.
So the first three multiples of 11 are: 11, 22, 33.
Pattern: for multiples of 11 up to 99, both digits are the same. ✓
Q19.
Find any common multiple of 4 and 6.
Your answer:(1 mark)
A common multiple of 4 and 6 is a number that appears in BOTH times tables.
List multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …
List multiples of 6: 6, 12, 18, 24, 30, 36, …
Find numbers that appear in both lists: 12, 24, 36, 48, …
Any of these is a correct answer. The smallest one (12) has a special name — the LCM. ✓
Q20.
Find the lowest common multiple of 3 and 7.
Your answer:(2 marks)
The LCM of 3 and 7 is the smallest number that is a multiple of both.
Step 1: Do 3 and 7 share any factors (other than 1)?
3 has factors {1, 3}. 7 has factors {1, 7}. They share only 1.
Step 2: When two numbers share no factors, their LCM is just their product.
LCM = 3 × 7 = 21.
Check: 21 is in 3’s table (3×7=21) and in 7’s table (7×3=21). ✓
Section B · Common multiples and word problems
Find common multiples and the LCM by listing. Apply to word problems.
LEARNING OBJECTIVE 2 · GRADE 1–3
Find common multiples and the lowest common multiple (LCM) by listing, and apply to word problems.
Success criteria — I can:
list multiples of two numbers and pick out the common ones
identify the LCM as the smallest common multiple
recognise a word problem that asks for an LCM (repeating events meeting again)
explain reasoning using divisibility rules
Q21.
Find all four multiples of 7 between 25 and 50.
Your answer:
,,,
(4 marks)
We need every multiple of 7 that sits between 25 and 50.
List multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, …
Pick the ones between 25 and 50: 28, 35, 42, 49.
(21 is below 25; 56 is above 50.) ✓
Q22.
Find all four multiples of 11 between 40 and 80.
Your answer:
,,,
(4 marks)
We need every multiple of 11 that sits between 40 and 80.
List multiples of 11: 11, 22, 33, 44, 55, 66, 77, 88, …
Pick the ones between 40 and 80: 44, 55, 66, 77.
(33 is below 40; 88 is above 80.) ✓
Q23.
Find all four multiples of 12 between 30 and 80.
Your answer:
,,,
(4 marks)
We need every multiple of 12 that sits between 30 and 80.
List multiples of 12: 12, 24, 36, 48, 60, 72, 84, …
Pick the ones between 30 and 80: 36, 48, 60, 72.
(24 is below 30; 84 is above 80.) ✓
Q24.
From the list 18, 24, 30, 36, 42, 48, find the two multiples of 8.
Your answer:
,
(2 marks)
We test each number to see if it’s a multiple of 8 (divide and check for a whole number).
18 ÷ 8 = 2.25. Not whole. NO.
24 ÷ 8 = 3. Whole. YES — 24 IS a multiple of 8.
30 ÷ 8 = 3.75. Not whole. NO.
36 ÷ 8 = 4.5. Not whole. NO.
42 ÷ 8 = 5.25. Not whole. NO.
48 ÷ 8 = 6. Whole. YES — 48 IS a multiple of 8.
Answer: 24 and 48. ✓
Q25.
Hassan says: “45 is a multiple of 6 because 45 is bigger than 6.” Is he correct?
Your answer:(2 marks)
Hassan’s claim: “a number that is bigger than 6 is automatically a multiple of 6.”
This is wrong. Being bigger than 6 does not put a number in the 6 times table.
Test 45: 45 ÷ 6 = 7.5. Not a whole number.
So 45 is NOT a multiple of 6.
The multiples of 6 near 45 are 42 (= 6×7) and 48 (= 6×8). 45 sits between them but isn’t one of them.
Hassan is incorrect. ✗
Q26.
Find the lowest common multiple of 6 and 9.
Your answer:(2 marks)
We need the smallest number that is a multiple of both 6 and 9 (the LCM).
List multiples of 6: 6, 12, 18, 24, 30, 36, …
List multiples of 9: 9, 18, 27, 36, …
Find the smallest number that appears in both lists: 18.
So LCM(6, 9) = 18. ✓
Watch out: 6 × 9 = 54 is also a common multiple, but NOT the smallest. 6 and 9 share the factor 3, so the multiply-shortcut overshoots.
Q27.
Find the lowest common multiple of 4 and 10.
Your answer:(2 marks)
We need LCM(4, 10).
List multiples of 4: 4, 8, 12, 16, 20, 24, …
List multiples of 10: 10, 20, 30, …
Smallest in both: 20.
So LCM(4, 10) = 20. ✓
(4 × 10 = 40 is a common multiple but not the smallest. 4 and 10 share the factor 2.)
Q28.
Find two common multiples of 3 and 5 that are both less than 50.
Your answer:
,
(2 marks)
A common multiple of 3 and 5 is a number in both times tables.
First find the LCM of 3 and 5. They share no factors, so LCM = 3 × 5 = 15.
So the common multiples of 3 and 5 are: 15, 30, 45, 60, 75, …
Pick the ones less than 50: 15, 30, 45.
Any two of these is a correct answer. ✓
Q29.
Sara says: “The LCM of two numbers is always equal to multiplying them together.” Is she correct?
Your answer:(2 marks)
Sara’s claim: “LCM(a, b) = a × b for any two numbers.”
This is wrong. The shortcut only works when a and b share NO factors (other than 1).
Counter-example: LCM(4, 6) = 12, NOT 4 × 6 = 24.
Why? 4 and 6 share the factor 2, so the product 24 overshoots — 12 is already a common multiple.
Sara is incorrect. ✗
Q30.
For which pair does the LCM equal their product?
Your answer:(2 marks)
LCM = product is only true for pairs that share no factors (other than 1).
(a) 4 and 6: both even, share factor 2 → LCM = 12, not 24. ✗
(b) 5 and 7: both prime, share no factors → LCM = 5 × 7 = 35. ✓
(c) 6 and 8: both even, share factor 2 → LCM = 24, not 48. ✗
(d) 4 and 8: 4 divides 8, share factor 4 → LCM = 8, not 32. ✗
Answer: (b) 5 and 7.
Q31.
One bell rings every 6 minutes. Another bell rings every 8 minutes. They both ring at 12:00. How many minutes until they next ring at the same time?
Your answer:(2 marks)
The bells next ring at the same time after a number of minutes equal to the LCM of 6 and 8.
List multiples of 6: 6, 12, 18, 24, …
List multiples of 8: 8, 16, 24, …
Smallest in both: 24.
So the next time both bells ring together is 24 minutes later, at 12:24. ✓
Q32.
Eggs are sold in packs of 6 only. Buns are sold in packs of 8 only. You want to buy the same number of eggs as buns, in whole packs only. What is the smallest number you can buy?
Your answer:(2 marks)
We need a number that is a whole number of egg packs AND a whole number of bun packs.
That means a number in the 6 times table AND the 8 times table — the LCM of 6 and 8.
Multiples of 6: 6, 12, 18, 24, …
Multiples of 8: 8, 16, 24, …
Smallest in both: 24.
So you buy 24 eggs (= 4 packs of 6) and 24 buns (= 3 packs of 8). ✓
Q33.
Why is 168 a multiple of 6?
Your answer:(2 marks)
The rule for ÷6: a number must be a multiple of BOTH 2 AND 3.
Check ÷2 on 168: last digit is 8 (even). ✓ So 168 is a multiple of 2.
Check ÷3 on 168: digit sum 1 + 6 + 8 = 15. 15 is a multiple of 3. ✓ So 168 is a multiple of 3.
Both rules pass, so 168 IS a multiple of 6.
Answer: “Because 168 is a multiple of 2 AND a multiple of 3.”
Watch out: “even” alone is not enough — 14 is even but not a multiple of 6 (because 14 fails the ÷3 test).
Q34.
Why is 245 NOT a multiple of 4?
Your answer:(2 marks)
The rule for ÷4: the last two digits must form a multiple of 4.
The last two digits of 245 are 45.
Is 45 a multiple of 4? 45 ÷ 4 = 11.25, not a whole number.
So 45 is NOT a multiple of 4.
Therefore 245 is NOT a multiple of 4.
Answer: “Because the last two digits (45) are not a multiple of 4.” ✗
Q35.
Which rule shows that 432 is a multiple of 9?
Your answer:(2 marks)
The rule for ÷9: the digit sum must be a multiple of 9.
Add the digits of 432: 4 + 3 + 2 = 9.
Is 9 a multiple of 9? Yes (9 × 1 = 9).
So 432 IS a multiple of 9 (in fact 432 = 9 × 48).
Answer: “The digit sum 4+3+2 = 9 is a multiple of 9.”
Watch out: being in the 3 times table doesn’t prove ÷9. Many multiples of 3 (like 6, 12, 15, 21) are NOT multiples of 9.
Section C · Cross-link calculation
Stretch: times tables, multiples in a range, and LCM with cross-arithmetic.
LEARNING OBJECTIVE 3 · GRADE 3
Combine multiples knowledge with times-table fluency in cross-topic calculations.
Success criteria — I can:
recall times-table facts to find multiples quickly
find all the multiples of a number within a given range
solve LCM problems where the answer must satisfy more than one condition
Q36.
? × 6 = 42. Find ?.
Your answer:(2 marks)
We need a number that, when multiplied by 6, gives 42.
Rearrange: ? = 42 ÷ 6.
42 ÷ 6 = 7.
Check: 7 × 6 = 42. ✓
(42 is the 7th multiple of 6.)
Q37.
Find a multiple of 8 between 60 and 70.
Your answer:(2 marks)
List multiples of 8 near 60–70: 8, 16, 24, 32, 40, 48, 56, 64, 72, …
The one between 60 and 70 is 64.
Check: 8 × 8 = 64. ✓
(56 is below 60; 72 is above 70.)
Q38.
7 × 9 = ?
Your answer:(1 mark)
7 × 9 = 63.
Memory tip: 7 × 9 = (7 × 10) − 7 = 70 − 7 = 63.
Digit-sum check: 6 + 3 = 9. ✓ (Consistent with 63 being a multiple of 9.)
Q39.
A number is a multiple of 4 AND a multiple of 9. What is the smallest such number?
Your answer:(2 marks)
We need the smallest number that is a multiple of 4 AND of 9 (the LCM).
Do 4 and 9 share any factors? 4 = 2 × 2. 9 = 3 × 3. They share no factors (other than 1).
So LCM = 4 × 9 = 36.
Check: 36 = 4 × 9 (multiple of 4) ✓, and 36 = 9 × 4 (multiple of 9) ✓.
Q40.
Find a multiple of 6 between 80 and 89.
Your answer:(2 marks)
List multiples of 6 near 80–89: 78, 84, 90, …
The one between 80 and 89 is 84.
Check: 6 × 14 = 84. Even (ends in 4) ✓, digit sum 8 + 4 = 12 (multiple of 3) ✓. So 84 is a multiple of 6. ✓
Section D · Cross-link word problems
Stretch: real-life problems where multiples and LCM appear.
LEARNING OBJECTIVE 4 · GRADE 3–4
Apply multiples and LCM in real-life cross-topic contexts.
Success criteria — I can:
spot when a real-life problem is asking for an LCM (events meeting, items combining)
set up and solve the LCM calculation by listing
check the answer makes sense in the real-life context
Q41.
A class has 30 pupils. The teacher wants to arrange them in rows of 6 pupils per row. Can she do this with no pupils left over?
Your answer:(2 marks)
We need to check if 30 can be split into equal rows of 6.
This is the same as asking: is 30 a multiple of 6?
30 ÷ 6 = 5 (a whole number).
So yes — the class forms 5 rows of 6 exactly. ✓
Q42.
A school can hire coaches that hold 12 pupils, or coaches that hold 18 pupils. They want a number of pupils that fills coaches of either size exactly — with no empty seats. What is the smallest such number?
Your answer:(3 marks)
We need a number that fills coaches of 12 exactly AND coaches of 18 exactly.
So the number must be a multiple of both 12 AND 18 — the LCM.
Multiples of 12: 12, 24, 36, 48, 60, …
Multiples of 18: 18, 36, 54, …
Smallest in both: 36.
So LCM(12, 18) = 36 pupils. (That’s 3 small coaches OR 2 big coaches.) ✓
Q43.
One clock chimes every 15 minutes. Another chimes every 20 minutes. They both chime at 9:00 am. How many minutes until they next chime together?
Your answer:(3 marks)
The next together-chime is at the LCM of 15 and 20 (in minutes).
Multiples of 15: 15, 30, 45, 60, …
Multiples of 20: 20, 40, 60, …
Smallest in both: 60.
So they next chime together 60 minutes later — at 10:00 am. ✓
Q44.
A box has 24 sweets. The teacher wants to share them equally between 5 pupils, so each pupil gets the same number. Can she?
Your answer:(2 marks)
To share 24 equally between 5 pupils, 24 must be a multiple of 5.
Test: 24 ÷ 5 = 4.8. NOT a whole number.
So 24 is NOT a multiple of 5.
She cannot share equally without breaking sweets. ✗
(The closest multiples of 5 are 20 (would leave 4 over) and 25 (would be 1 short).)
Q45.
Stefon is making cheese straws. He wants to share them equally between 4, 5, OR 6 people (one of those, depending on who turns up). What is the smallest number he should make?
Your answer:(3 marks)
Stefon needs a number that splits equally among 4, 5, OR 6 people.
So the number must be a multiple of 4 AND 5 AND 6 — the LCM of all three.
Step 1: Find LCM(4, 5). They share no factors, so LCM = 4 × 5 = 20.
Step 2: Now find LCM(20, 6). Multiples of 20: 20, 40, 60, 80, … Is 60 a multiple of 6? Yes (60 = 6 × 10).
So LCM(4, 5, 6) = 60.
Check: 60 ÷ 4 = 15 each (4 people). 60 ÷ 5 = 12 each (5 people). 60 ÷ 6 = 10 each (6 people). All whole. ✓
Your result
Your answers are now locked. Scroll up to find your wrong answers — each one has the correct answer, full working, and 5 similar practice questions below it.
Sending to tutor
✉
1. Main report
Click “Send to Tutor” below to open the email.
✓
2. Progress update (after extra practice)
Answer practice questions under your wrong answers, then send an update showing your work.
Both messages send to gulamhusain.dabhad@gmail.com and info@improvetuition.org, with copies to you and your named tutor.