3.1 Factors, multiples and divisibility — Learn

Prerequisites: multiplication tables, and dividing one whole number by another.

Objectives: list the factors and multiples of a number; write a number as a product of two factors; test divisibility by 2, 3, 4, 5, 6, 9 and 10; recognise square and cube numbers.

By the end of this section, you can find every factor of a number, generate multiples of it, and say whether one number divides another without doing the division.

Factors

A factor of a number divides it exactly, leaving no remainder. The factors of 12 are the whole numbers you can multiply by something to get 12.

So the factors of 12 are $1, 2, 3, 4, 6$ and $12$.

Notice how they pair up. Each pair multiplies to 12, and the two numbers in a pair sit at opposite ends of the list: 1 goes with 12, 2 with 6, 3 with 4.

Example 1. Write down all the factors of 40.

The factors are $1, 2, 4, 5, 8, 10, 20$ and $40$.

Working in pairs is what makes this quick. You start from 1 and go up, and you stop once you meet a number you have already used — that is, once you reach the square root. For 40 you stop after 5, because $6^2=36$ is past 40 and 7 and 8 are already partners of numbers you have written down. Writing the pairs as pairs also means you cannot miss one.

Example 2. Write down all the factors of 36.

The factors are $1, 2, 3, 4, 6, 9, 12, 18$ and $36$. There are nine of them, which is an odd number of factors. That happens because 36 is a square number: the factor 6 pairs with itself rather than with a different number.

Watch out. A factor is not the same as a prime factor. The factors of 12 include 4 and 6, which are not prime. Prime factors are a chapter section of their own, and they are the ones that cannot be broken down any further.

Multiples

A multiple of a number is what you get when you multiply it by a whole number. The multiples of 7 are 7, 14, 21, 28, 35, and so on forever.

Every number is a multiple of itself, and every number is a multiple of 1. The list never ends, so a question will always say how many to write down or where to stop.

Example 3. Write down the multiples of 8 that are less than 60.

The next one is 64, which is past 60, so you stop there.

The words are easy to confuse, so keep the two sentences apart:

• $3$ is a factor of $12$, because $12\div3=4$ exactly.
• $12$ is a multiple of $3$, because $12=3\times4$.

Divisibility rules

You do not always need to divide to know whether something divides. These rules work because of how our numbers are written in tens.

Example 4. Is 4512 divisible by 6?

It ends in 2, so it is even and divides by 2.

Its digit sum is $4+5+1+2=12$, and 12 is a multiple of 3, so it divides by 3.

It divides by 2 and by 3, so it divides by 6. Yes.

Example 5. Is 1234 divisible by 3?

Its digit sum is $1+2+3+4=10$. That is not a multiple of 3, so 1234 is not divisible by 3 — and so it is not divisible by 6 either, even though it is even.

Watch out. The rule for 3 is about the sum of the digits, not the last digit. 1234 ends in 4 and is even, so it divides by 2; but it does not divide by 3, and the last digit tells you nothing about that. Use the digit sum.

Square and cube numbers

A square number is a number times itself: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$.

A cube number is a number times itself times itself: $1, 8, 27, 64, 125$.

Square numbers have an odd number of factors, for the reason you met above. Every other number has an even number.

Progress check

Earlier in this chapter you read that two lighthouses flash every 12 seconds and every 18 seconds, and that you would be able to work out when they next flash together. Before you do any arithmetic, decide which of these two ideas the question needs:

• the numbers that divide both 12 and 18;
• the numbers that are multiples of both 12 and 18.

Write down your choice and one sentence saying why. You will use it in Section 3.3.

You should now be able to

• list every factor of a number, working in pairs and stopping at the square root;
• write down multiples of a number to a given point;
• test divisibility by 2, 3, 4, 5, 6, 9 and 10 and say which rule you used;
• tell a factor from a multiple, and a square number from a cube number.