Divisibility Tests

Syllabus tag: KCSE | Mathematics | Form 1 | Topic 3 Divisibility Tests

Lesson objectives

By the end of this topic, you should be able to:

• Test the divisibility of numbers by 2, 3, 4, 5, 6, 8, 9, 10 and 11.
• Apply divisibility tests to simplify prime factorisation.
• Use combined tests for composite divisors.
• Explain why the tests for 3 and 9 work.

Divisibility Tests

A divisibility test tells you whether one number divides into another exactly, without carrying out the division.

a) The tests

Divisor	Test
2	the last digit is even
3	the digits add to a multiple of 3
4	the last two digits form a multiple of 4
5	the last digit is 0 or 5
6	it passes the tests for 2 and for 3
8	the last three digits form a multiple of 8
9	the digits add to a multiple of 9
10	the last digit is 0
11	alternately add and subtract the digits; the result is 0 or a multiple of 11

b) Using them

Take 4 716.

It is even, so 2 divides it.

Its digits add to 4 + 7 + 1 + 6 = 18. That is a multiple of 3 and of 9, so both divide it.

It passes both 2 and 3, so 6 divides it.

Its last two digits, 16, make a multiple of 4. So 4 divides it.

c) Combined tests

For a composite divisor, test its prime factors instead.

Since 6 = 2 × 3, a number divisible by both 2 and 3 is divisible by 6.

Since 12 = 4 × 3, test for 4 and for 3.

The factors used must have no common factor themselves. Testing 12 as 6 × 2 does not work, because 6 and 2 share a factor of 2.

d) Why the tests for 3 and 9 work

Notice that 10 is 9 + 1. Also 100 is 99 + 1, and 1 000 is 999 + 1.

Every power of ten is one more than a multiple of 9.

Split a number into its digits. Each digit contributes a multiple of 9, plus the digit itself.

The multiples of 9 divide out. What is left over is the sum of the digits.

That is why adding the digits tells you the remainder on dividing by 9. And since 9 is a multiple of 3, the same test works for 3.

e) The test for 11

Take 8 291.

Add alternate digits: 8 + 9 = 17. Add the others: 2 + 1 = 3.

Subtract: 17 − 3 = 14. Since 14 is not a multiple of 11, 8 291 is not divisible by 11.

f) Using tests in factorisation

Divisibility tests make prime factorisation much faster.

Instead of guessing, test 2, then 3, then 5, then 7, and so on.

For 594: it is even, so divide by 2 to get 297. Its digits add to 18, so 3 divides it, giving 99. Then 99 = 9 × 11. So 594 = 2 × 3³ × 11.

g) Where this is used

Prime factorisation. Simplifying fractions quickly. Checking arithmetic. Spotting whether a number of items can be shared equally.

Words to know

• Divisible -- dividing exactly, with no remainder.
• Divisibility test -- a rule for checking divisibility without doing the division.
• Digit sum -- the total obtained by adding all the digits of a number.
• Alternating sum -- the result of adding and subtracting digits in turn.
• Composite divisor -- a divisor that is not prime, such as 6 or 12.

:::checkpoint Check yourself

1. Is 3 528 divisible by 8? Show the test.
2. Is 2 079 divisible by 9? Show the test.
3. Why does testing 12 as 6 × 2 not work?
4. Explain why adding the digits tests for divisibility by 9. :::

Bridge to practice

The exercises begin with the last-digit and digit-sum tests, move through the test for 11 and combined tests, and finish with why the rules work and where they run out. When factorising, apply the tests in ascending order — 2, 3, 5, 7, 11 — rather than guessing which prime to try.