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You will learn: Explain divisibility tests and use prime factors to decide what divides a number.

Before you start: Multiplication, division and place value.

What does “divides” mean?

A positive integer d divides an integer n when n = dk for some integer k. We write d ∣ n. For example, 6 ∣ 42 because 42 = 6 × 7, whereas 6 does not divide 44. Every positive integer divides 0 because 0 = d × 0.

A prime is an integer greater than 1 whose only positive divisors are 1 and itself. Thus 2, 3, 5 and 7 are prime; 1 is not prime. An integer greater than 1 that is not prime is composite.

Build numbers from primes

Repeatedly split a composite factor until every factor is prime. For example, 84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7. The notation 2² means two factors equal to 2, not 2 × 2 factors of some other number.

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7
Stop splitting when all the factors are prime.

A divisor must use no more copies of each prime than the original number contains. Thus 12 = 2² × 3 divides 84, but 8 = 2³ does not: 84 has only two factors of 2. For positive integers greater than 1, prime factorisation is unique apart from the order of the factors.

Worked example 1: why the digit test works

Is 4,725 divisible by 9? Its digit sum is 4 + 7 + 2 + 5 = 18, which is divisible by 9. To understand the test, write 4,725 = 4 × 1,000 + 7 × 100 + 2 × 10 + 5. Each of 1,000, 100 and 10 is one more than a multiple of 9. Subtracting the digit sum therefore leaves a multiple of 9. The original number is divisible by 9 exactly when its digit sum is. The same reasoning works with 3.

Worked example 2: a missing digit

Find the digit x that makes 53x divisible by 9. Here 53x means the three-digit number with digits 5, 3, x, not a product. Its digit sum is 8 + x. For 0 ≤ x ≤ 9 this lies between 8 and 17. The only multiple of 9 in that interval is 9, so x = 1. Check: 531 = 9 × 59.

Combining divisibility facts

Divisibility by both 2 and 3 guarantees divisibility by 6 because these tests supply separate prime factors. Divisibility by 4 and 6 does not guarantee divisibility by 24: both requirements already contain a factor of 2. For instance 12 is divisible by 4 and 6, but not by 24. Prime factors help you avoid counting the same factor twice.

Practise at your next step

A six-question session chooses from nine questions. Two correct answers in a row without hints move you up a level; an incorrect answer brings a simpler next question where one is available. Hints keep you at the same level. This is a practice suggestion, not an exam score or proof of mastery.

Interactive practice loads here. You can also use the complete question set below.

Write a proof of your own

Prove that if a positive integer n is divisible by both 4 and 9, then it is divisible by 36.

Write your reasoning on paper before comparing. The practice checker does not grade a written proof.

Compare your proof with a full solution

Since 4 divides n, its prime factorisation contains at least two copies of 2. Since 9 divides n, it contains at least two copies of 3. These are different primes, so all four required factors occur together. Their product is 2² × 3² = 36. Hence 36 divides n.

Check: did you state the assumptions, explain the key step, and reach the requested conclusion?

All nine practice questions, hints and solutions

This complete set works without the interactive practice. Hide each solution until you have made an attempt.

1. Which of these integers is prime?

Foundation

  1. 1
  2. 21
  3. 29
  4. 39
Hint

If two integer factors are both at least 6, their product is at least 36.

Answer and explanation

29. If 29 were composite, it would have two integer factors both at least 2. They cannot both be at least 6, since 6 × 6 = 36 > 29. Thus one factor would be 2, 3, 4 or 5, but none divides 29. Therefore 29 is prime. Also 21 = 3 × 7, 39 = 3 × 13, and 1 is not prime.

2. What is the prime factorisation of 60?

Foundation

  1. 2² × 3 × 5
  2. 2 × 3² × 5
  3. 2² × 15²
  4. 3 × 20
Hint

Split 60 into 6 × 10, then split each factor into primes.

Answer and explanation

2² × 3 × 5. 60 = 6 × 10 = (2 × 3)(2 × 5) = 2² × 3 × 5. Although 3 × 20 equals 60, it is not a prime factorisation because 20 is composite.

3. Which number is divisible by 9?

Foundation

  1. 234
  2. 235
  3. 236
  4. 238
Hint

Compare the sums of the digits.

Answer and explanation

234. The digit sum of 234 is 2 + 3 + 4 = 9, so 234 is divisible by 9. The other digit sums are 10, 11 and 13, none a multiple of 9.

4. The notation 71x represents a three-digit number. Which digit x makes it divisible by 9?

Core

  1. 0
  2. 1
  3. 2
  4. 9
Hint

The digit sum is 8 + x.

Answer and explanation

1. For a digit x, the sum 8 + x ranges from 8 to 17. Its only possible multiple of 9 is 9, giving x = 1. Indeed 711 = 9 × 79.

5. Which condition guarantees that an integer is divisible by 6?

Core

  1. Divisible by 2 only
  2. Divisible by 3 only
  3. Divisible by both 2 and 3
  4. Ends in 6
Hint

The prime factors of 6 are 2 and 3.

Answer and explanation

Divisible by both 2 and 3. Divisibility by 2 and 3 supplies both distinct prime factors, so 6 divides the integer. Neither condition alone is enough, and 16 ends in 6 but is not divisible by 6.

6. Which proposed divisor does NOT divide 2³ × 3²?

Core

  1. 8
  2. 9
  3. 12
  4. 16
Hint

Compare the number of factors equal to 2.

Answer and explanation

16. The number has three factors of 2. A divisor 16 = 2⁴ would need four, so 16 cannot divide it. The factors needed for 8, 9 and 12 are all available.

7. What is the smallest positive integer divisible by both 8 and 12?

Stretch

  1. 12
  2. 24
  3. 48
  4. 96
Hint

8 needs three factors of 2; 12 needs two factors of 2 and one of 3.

Answer and explanation

24. Use the largest required count for each prime: 2³ × 3 = 24. This is divisible by 8 and by 12, and any common multiple must contain these prime factors, so no smaller positive one works.

8. If a positive integer is divisible by 6 and 10, what must divide it?

Stretch

  1. 16
  2. 30
  3. 60
  4. 100
Hint

List the prime factors required by each condition.

Answer and explanation

30. 6 needs 2 and 3; 10 needs 2 and 5. Together they require 2 × 3 × 5 = 30. A second factor of 2 is not guaranteed: 30 itself meets the assumptions but is not divisible by 60.

9. How many positive divisors does 18 = 2 × 3² have?

Stretch

  1. 4
  2. 5
  3. 6
  4. 8
Hint

A divisor may use zero or one factor of 2, and zero, one or two factors of 3.

Answer and explanation

6. There are 2 choices for the exponent of 2 and 3 choices for the exponent of 3. Each pair gives a different divisor, so there are 2 × 3 = 6. They are 1, 2, 3, 6, 9 and 18.

Where to go next

Move on to the mixed proof-practice plan · Explore the wider Olympiad topic map

Teaching examples prepared for this guide, not attributed to a contest paper. Familiar elementary problems may appear in other learning materials. Report an unclear step or an error.