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

Work independently. The questions and answers are shown together at the end.

Several of these are word problems. Before you calculate, decide whether the question wants the highest common factor or the lowest common multiple, and write that decision down. A correct number from the wrong idea scores nothing.

Find the highest common factor of 24 and 36.
Find the lowest common multiple of 24 and 36.
Find the highest common factor of 84 and 210.
Find the lowest common multiple of 12, 15 and 20.
Two lighthouses flash every 12 seconds and every 18 seconds. They flash together. After how many seconds do they next flash together?
A shop has 48 blue pens and 60 black pens. They are packed into boxes of one colour, every box holding the same number of pens, with none left over. What is the greatest number of pens a box can hold?
H For any two whole numbers, the product of their highest common factor and their lowest common multiple is the product of the numbers. Check this for 18 and 30.

Answers and concise solutions

1. Find the highest common factor of 24 and 36.

Answer: $12$ . In prime factors, $24=2^3\times3$ and $36=2^2\times3^2$ ; the highest common factor takes the smaller power of each shared prime, $2^2\times3=12$ .

2. Find the lowest common multiple of 24 and 36.

Answer: $72$ . The lowest common multiple takes the larger power of each shared prime, $2^3\times3^2=72$ .

3. Find the highest common factor of 84 and 210.

Answer: $42$ . $84=2^2\times3\times7$ and $210=2\times3\times5\times7$ , so the highest common factor is $2\times3\times7=42$ .

4. Find the lowest common multiple of 12, 15 and 20.

Answer: $60$ . In prime factors: $12=2^2\times3$ , $15=3\times5$ , $20=2^2\times5$ . The lowest common multiple needs the largest power of each prime that appears, $2^2\times3\times5=60$ .

5. After how many seconds do they next flash together?

Answer: $36$ seconds. The question asks when two repeating events line up again, which is the lowest common multiple — not the highest common factor.

6. What is the greatest number of pens a box can hold?

Answer: $12$ pens. The box size must divide both 48 and 60, and it must be as large as possible, so it is the highest common factor.

7. Check the rule for 18 and 30.

Answer: For 18 and 30: the highest common factor is 6 and the lowest common multiple is 90, and $6\times90=540=18\times30$ . The rule holds, which is why finding one of the two gives you the other.