3.3 The highest common factor and the lowest common multiple — Practise with help

Warm-up: remember

Write 36 as a product of its prime factors.
Write 60 as a product of its prime factors.

Answers and feedback

$36=2^2\times3^2$ .
$60=2^2\times3\times5$ .

You will need both of these in this section. Taking the smaller power of each shared prime gives one answer, and the larger power gives the other.

Practise with help

1. Find the highest common factor of 12 and 18.

Hint

Write both in prime factors: $12=2^2\times3$ and $18=2\times3^2$ . Which power of each shared prime do you take?

Next step

The smaller: $2^1$ and $3^1$ .

Full working

\text{HCF} = 2\times3 = 6

Check: 6 divides 12 (twice) and divides 18 (three times), and nothing larger does — 8 divides 12 but not 18, and 9 divides 18 but not 12.

The highest common factor cannot be bigger than 12, the smaller number, and 6 is not bigger. That is the right size.

2. Find the lowest common multiple of 12 and 18.

Hint

Same two factorisations, but take the larger power of each prime.

Next step

$2^2$ and $3^2$ .

Full working

\text{LCM} = 2^2\times3^2 = 36

Check: $36\div12=3$ and $36\div18=2$ , so 36 is a multiple of both.

The lowest common multiple cannot be smaller than 18, the larger number, and 36 is not smaller.

3. Find the highest common factor of 36 and 60.

Hint

From the warm-up: $36=2^2\times3^2$ and $60=2^2\times3\times5$ . Both contain 2 and 3. Does either contain 5?

Next step

5 appears only in 60, so it cannot be in a common factor.

Full working

\text{HCF} = 2^2\times3 = 12

Check: $36\div12=3$ and $60\div12=5$ . Both are whole numbers, so 12 divides both.

4. Find the lowest common multiple of 36 and 60.

Hint

Now you need every prime that appears in either number.

Next step

That is 2, 3 and 5 — with the largest power each one reaches.

Full working

\text{LCM} = 2^2\times3^2\times5 = 180

Check: $180\div36=5$ and $180\div60=3$ . Both whole, so 180 is a common multiple. And 36 is not a multiple of 60, nor 60 of 36, so nothing between 60 and 180 works.

5. Buses leave a stop every 15 minutes and every 20 minutes. They leave together at 9:00. When do they next leave together?

Hint

Two repeating events lining up. Which of the two ideas does that need?

Next step

The lowest common multiple. $15=3\times5$ and $20=2^2\times5$ .

Full working

\text{LCM} = 2^2\times3\times5 = 60

So they next leave together 60 minutes later, at 10:00.

Notice the answer is bigger than both 15 and 20, which is what should happen for a lowest common multiple.

6. Ribbon comes in 24 cm and 90 cm lengths. It is cut into equal pieces with none wasted, and each piece is as long as possible. How long is each piece?

Hint

The piece length must divide both 24 and 90, and be as large as possible.

Next step

That is the highest common factor. $24=2^3\times3$ and $90=2\times3^2\times5$ .

Full working

\text{HCF} = 2\times3 = 6

Each piece is 6 cm. Then the 24 cm ribbon gives 4 pieces and the 90 cm gives 15.

The answer is smaller than both numbers, which is what should happen for a highest common factor.

7. A question gives two numbers and asks for their highest common factor. The answer you calculate is bigger than both numbers. What has gone wrong?

Hint

What is the biggest a common factor of two numbers can possibly be?

Next step

It cannot be bigger than the smaller number.

Full working

You have found the lowest common multiple instead. That is a multiple, so it is at least as big as the larger number.

Go back to the table: highest common factor takes the smaller power of each shared prime, and the lowest common multiple takes the larger. If you took the larger powers and the answer is too big, that is why.

This check is worth doing every time, because it costs nothing and catches the mistake on the spot.

8. Two timers beep every 8 seconds and every 14 seconds. A third beeps every 20 seconds. All three beep together. How long until all three beep together again?

Hint

Three numbers this time, but the idea is the same. Work out the lowest common multiple of two of them first, then of that answer and the third.

Next step

$8=2^3$ , $14=2\times7$ , $20=2^2\times5$ . You need the largest power of every prime that appears anywhere.

Full working

\text{LCM} = 2^3\times5\times7 = 280

So after 280 seconds.

Doing it in two steps gives the same answer: $8$ and $14$ have a lowest common multiple of $56$ , and the lowest common multiple of $56$ and $20$ is $280$ . That is a good way to handle three numbers without losing track.

A question from Chapter 1

9. Work out $(-3)\times(-8)$ .

Hint

Multiplying two negative numbers: is the answer positive or negative?

Next step

Two negatives make a positive.

Full working

$24$ .

$3\times8=24$ , and because both numbers are negative the answer is positive.

It is the same answer as $3\times8$ , which is why 24 has the factors it has: $-3$ and $-8$ also multiply to 24. Factors of a number are usually taken to be positive unless the question says otherwise.