Factors, Multiples and Divisibility | Mathematics

Unit 1 · Integers

1.2 Factors, Multiples and Divisibility

Two bells ringing at different intervals, two library visitors on different schedules, two friends building identical bouquets — they're all exactly the same question in disguise.

🎯 By the end of this you can…

✓Find the LCM and HCF of two numbers using prime factors
✓Test divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11 without dividing
✓Build a Sieve of Eratosthenes to find every prime up to 30
✓Recognise square and cube numbers, including which numbers are both
✓Reduce a fraction to its simplest form using the HCF

🕸️ Athena's Sieve — click the next prime yourself

Click the smallest number still plain — if nothing smaller crossed it out, it's prime.

💡 The idea

Why the sieve works: the next uncrossed number can only be prime

You just circled each prime yourself. Every time you picked the SMALLEST number still untouched, nothing smaller could have divided it — if it had a smaller factor, that factor would already have crossed it out earlier. That's the entire proof, not just a rule to memorise.

🔢 Test any number's divisibility — type one in

Isdivisible by...

2YES

3YES

4no

5no

6YES

8no

9YES

10no

2: last digit 8 is even.

3: digit sum 18 is a multiple of 3.

4: last two digits (58) don't divide by 4.

5: last digit is 8.

6: divisible by both 2 AND 3.

8: last three digits (158) don't divide by 8.

9: digit sum 18 is a multiple of 9.

10: last digit is 8.

✅ 4158 is divisible by: 2, 3, 6, 9

💡 The idea

Divisibility tests check a number WITHOUT dividing it

Try your own numbers in the box above. Each test looks at a different part of the number: its last digit (2, 5, 10), its last two or three digits (4, 8), or the sum of all its digits (3, 9). Divisibility by 6 is just 2 AND 3 both passing.

⚠️ The classic trap

Eva's test for 3 works by adding ALL the digits, not just checking the last one. $1 + 5 + 4 = 10$ , and since 10 isn't a multiple of 3, 154 isn't divisible by 3 — even though the LAST digit, 4, has nothing to do with the test for 3.

💡 The idea

A ninth test: divisibility by 11, with ALTERNATING signs

Starting from the units digit, add and subtract digits alternately: $+$ , then $-$ , then $+$ , and so on. If that total is a multiple of 11 (including 0), the whole number is too.

🔢 Test any number's divisibility — type one in

Isdivisible by...

2YES

3YES

4no

5YES

6YES

8no

9YES

10YES

11no

2: last digit 0 is even.

3: digit sum 9 is a multiple of 3.

4: last two digits (30) don't divide by 4.

5: last digit is 0.

6: divisible by both 2 AND 3.

8: last three digits (630) don't divide by 8.

9: digit sum 9 is a multiple of 9.

10: last digit is 0.

11: alternating digit sum (right to left) is 3, not a multiple of 11.

✅ 630 is divisible by: 2, 3, 5, 6, 9, 10

✏️ Worked example

Which numbers below 12 are factors of 630?

1
$630 = 2 \times 3^{2} \times 5 \times 7$ .
2
2 ✓ (one factor of 2), 3 ✓ and 9 ✓ (has $3^{2}$ ), 5 ✓, 6 ✓ (2 and 3 both present), 10 ✓ (2 and 5 both present).
3
4 ✗ (needs $2^{2}$ , only has $2^{1}$ ), 8 ✗ (needs $2^{3}$ ), 11 ✗ (11 isn't among 630's prime factors — confirmed by the alternating-sum test too).

✅ Answer: 2, 3, 5, 6, 9 and 10 are factors of 630; 4, 8 and 11 are not.

💡 The idea

LCM and HCF both come from the same prime factor tree

The Lowest Common Multiple is the smallest number both numbers divide into. The Highest Common Factor is the biggest number that divides into both. Both are read straight off each number's prime factors — HCF takes the SHARED primes (lowest power each), LCM takes EVERY prime that appears (highest power each).

✏️ Worked example

Find the LCM and HCF of 24 and 40.

1
$24 = 2^{3} \times 3$ and $40 = 2^{3} \times 5$ .
2
HCF: the shared prime is $2^{3}$ only (3 and 5 don't both appear) $= 8$ .
3
LCM: every prime at its highest power — $2^{3} \times 3 \times 5 = 120$ .

✅ Answer: $HCF = 8, LCM = 120$

💡 The idea

Simplifying a fraction is just dividing both terms by their HCF

$\frac{48}{60}$ : HCF(48,60) = 12, so divide both terms by 12 — $\frac{48 \div 12}{60 \div 12} = \frac{4}{5}$ . That's as far as it reduces, because 4 and 5 share no common factor beyond 1.

🎮 LCM, HCF, fractions and divisibility — your turn

⭐ 0 XP🔥 streak 0

1Basic · 10XP

LCM of 6 and 10

2Basic · 10XP

HCF of 54 and 90

3Basic · 10XP

Write down the factors of 45

4Basic · 10XP

Write down the factors of 32

5Branching · 20XP

LCM of 12 and 15

6Branching · 20XP

LCM of 25 and 40

7Branching · 20XP

LCM of 32, 48 and 64

8Branching · 20XP

HCF of 96 and 72

9Branching · 20XP

Write down the common factors of 60 and 24

10Branching · 20XP

Find the common factors of 12, 36 and 60

11Branching · 20XP

Simplify

\frac{48}{60}

to its simplest form

12Branching · 20XP

Anna visits the library every 4 days, Ben every 6 days. Both went today. In how many days will they next both visit on the same day?

13Branching · 20XP

A florist has 24 red roses and 36 white roses, and wants identical bouquets using every rose with none left over. What's the maximum number of bouquets?

14Challenge · 35XP

The HCF of two numbers is 3. Their LCM is 18. Give one valid pair of numbers (there's more than one correct answer).

15Challenge · 35XP

A bell rings every 6 minutes, a light flashes every 8 minutes. Both happen together at 9:00am. At what time do they NEXT happen together?

16Challenge · 35XP

LCM of 20 and 21

17Challenge · 35XP

LCM of 7 and 3

18Challenge · 35XP

LCM of 12 and 36

19Challenge · 35XP

HCF of 234 and 236

20Challenge · 35XP

HCF of 2 and 13

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💡 The idea

Square and cube numbers — and numbers that are both

A square number is $n \times n$ ; a cube number is $n \times n \times n$ . A number can be both — 64 is $8^{2}$ AND $4^{3}$ , because $64 = 2^{6}$ and 6 is divisible by both 2 and 3.

✏️ Worked example

Why is 27 a cube number but not a square number?

1
$27 = 3^{3}$ , so it's a cube.
2
For 27 to be a square, some whole number multiplied by itself would have to give 27. $5^{2} = 25$ and $6^{2} = 36$ — 27 falls strictly between them, so no whole number squares to 27.

✅ Answer: 27 is a cube (3³) but not a square — it sits between 5² and 6².

🤔 Angelique says: 'Between 10 and 90, there are 6 more square numbers than cube numbers.' Is she correct?

🎮 Primes, squares and cubes

⭐ 0 XP🔥 streak 0

1Basic · 10XP

Is 29 a cube number?

2Basic · 10XP

Find a two-digit number that is both a square AND a cube number.

3Branching · 20XP

Using the pattern 2²=4, 4²=16, 8²=64, 16²=256, find 32².

4Branching · 20XP

Work out

144 + 64 + 25 + 81 + 16

5Branching · 20XP

Work out

4^{2} + 3^{2}

6Branching · 20XP

Work out

2 5^{2}

7Branching · 20XP

Work out

9^{2}

8Branching · 20XP

Work out

1^{0} + 2^{0} + 3^{0}

9Challenge · 35XP

Work out

364 - 64

10Challenge · 35XP

3*58 has its * digit replaced. The number is divisible by 3. List all possible values of *, separated by commas.

11Challenge · 35XP

HCF(501, 503)

12Challenge · 35XP

HCF(54321, 54322)

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You finished this lesson!

Prime factors unlock LCM, HCF, divisibility, and square/cube recognition all at once — four Unit 1 skills, one underlying idea. The Sentry is waiting.

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