3.1 Factors, multiples and divisibility — Practise with help

Warm-up: remember

Work out $7\times8$ .
Work out $5^2$ and $2^3$ .

Answers and feedback

$56$ . If you used $7\times7=49$ and added one more 7, that also works.
$5^2=25$ , because $5\times5=25$ . And $2^3=8$ , because $2\times2\times2=8$ . The index tells you how many factors, not what to multiply the base by.

Practise with help

1. Write down all the factors of 18.

Hint

Start at 1 and work up in pairs. Each pair multiplies to 18.

Next step

$18=1\times18=2\times9=3\times6$ .

Full working

The factors are $1, 2, 3, 6, 9$ and $18$ .

You can stop once you reach the square root. $\sqrt{18}$ is about 4.2, so you only needed to try 1, 2, 3 and 4. The 4 does not divide 18, and 5 is past the square root, so the list is complete.

2. Write down the first five multiples of 9.

Hint

A multiple of 9 is 9 times a whole number. Start at 9 times 1.

Next step

$9\times1$ , $9\times2$ , $9\times3$ , $9\times4$ , $9\times5$ .

Full working

9, 18, 27, 36, 45

Notice that the digit sums are $9, 9, 9, 9, 9$ — that is the divisibility rule for 9 appearing in the answer.

3. Is 372 divisible by 3? Give a reason.

Hint

Add the digits. Do not look at the last digit.

Next step

$3+7+2=12$ .

Full working

Yes. The digit sum is $3+7+2=12$ , and 12 is a multiple of 3. So 372 is divisible by 3.

You can check: $372\div3=124$ exactly.

4. Is 372 divisible by 6? Give a reason.

Hint

For 6 you need two things to be true. What are they?

Next step

A number divides by 6 when it divides by 2 and by 3.

Full working

Yes. It ends in 2, so it is even and divides by 2. From question 3, its digit sum is 12, so it divides by 3. Dividing by both 2 and 3 means it divides by 6.

Check: $372\div6=62$ .

5. Write down all the factors of 64.

Hint

64 is a square number, so one of its factor pairs is a number paired with itself.

Next step

$64=1\times64=2\times32=4\times16=8\times8$ .

Full working

The factors are $1, 2, 4, 8, 16, 32$ and $64$ .

There are seven — an odd number, because $8\times8=64$ pairs 8 with itself. Every square number has an odd number of factors, and every other number has an even number.

6. A number is divisible by 4 when its last two digits form a multiple of 4. Use this to decide whether 72516 is divisible by 4.

Hint

Look only at the last two digits: 16.

Next step

$16\div4=4$ exactly.

Full working

Yes. The last two digits are 16, and $16=4\times4$ , so 72516 is divisible by 4.

Check: $72516\div4=18129$ .

The rule works because 100 is a multiple of 4, so the hundreds and beyond never affect the remainder.

7. Write down the square numbers between 50 and 100, and say how many there are.

Hint

Which whole numbers squared land between 50 and 100?

Next step

$7^2=49$ is too small. Try $8^2$ , $9^2$ and $10^2$ .

Full working

$8^2=64$ , $9^2=81$ , and $10^2=100$ is not between 50 and 100 because it is equal to the upper end.

So there are two: $64$ and $81$ .

8. A teacher has 30 stickers and wants to give every pupil the same number, with none left over. Which numbers of pupils is this possible for?

Hint

The number of pupils must divide 30 exactly. So you need the factors of 30.

Next step

$30=1\times30=2\times15=3\times10=5\times6$ .

Full working

The pupils could be $1, 2, 3, 5, 6, 10, 15$ or $30$ in number — one for each factor of 30.

Each of those gives the same number of stickers per pupil: 30, 15, 10, 6, 5, 3, 2 or 1 respectively.

A question from an earlier section

9. Round 3174 to the nearest hundred.

Hint

Find the two hundreds it sits between: 3100 and 3200. Which is it closer to?

Next step

74 is nearer to 100 than to 0.

Full working

$3200$ . The tens digit is 7, which is 5 or more, so you round up from 3100 to 3200.

This is the kind of check worth doing before you trust an answer: $3174\div4$ is about $3200\div4$ , which is 800.