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

Prerequisites: listing factors and multiples, and writing a number as a product of primes.

Objectives: find the highest common factor and the lowest common multiple of two or more numbers; decide which of the two a word problem is asking for.

By the end of this section, you can answer the chapter problem and say why it needed the idea it needed.

The highest common factor

The highest common factor of two numbers is the largest number that divides both of them exactly.

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

In prime factors:

A common factor can only use primes that appear in both. And it cannot use more copies of a prime than the smaller of the two has. So:

• 2 appears in both. The smaller power is $2^2$, so take $2^2$.
• 3 appears in both. The smaller power is $3^1$, so take $3$.

The lowest common multiple

The lowest common multiple of two numbers is the smallest number that both divide into exactly.

Example 2. Find the lowest common multiple of 24 and 36, using the same factorisations.

A common multiple has to contain enough of each prime to build either number. So it needs the larger power of each prime that appears:

• 2: the larger power is $2^3$.
• 3: the larger power is $3^2$.

The two rules are the same rule with the comparison flipped.

Example 3. Find the highest common factor and the lowest common multiple of 84 and 210.

Both contain 2, 3 and 7. Only 210 contains 5.

Watch out. The highest common factor is never bigger than the smaller of the two numbers, and the lowest common multiple is never smaller than the larger of them. Those two facts are free checks. If you are asked for the highest common factor of 84 and 210 and you get 420, you have found the lowest common multiple — or you have used the larger powers where you needed the smaller ones.

Which one does the question want?

This is the part that decides the mark. Read what the question is doing with the numbers.

Sharing or dividing into equal groups means the highest common factor. The answer has to divide both numbers, so it is a factor.

Repeating events lining up means the lowest common multiple. The answer has to be a multiple of both, so it is a multiple.

Example 4. Two lighthouses flash every 12 seconds and every 18 seconds. They flash together. How long until they next flash together?

The question asks when two repeating events line up again. That is a multiple of both 12 and 18, and "next" makes it the lowest one.

They next flash together after 36 seconds.

Example 5. A shop has 48 blue pens and 60 black pens. They go into boxes of one colour, every box holds the same number, and nothing is left over. What is the greatest number of pens a box can hold?

The box size must divide 48 and it must divide 60. It must also be as large as possible, so it is the highest common factor.

A box can hold 12 pens. Then there are 4 boxes of blue pens and 5 boxes of black pens.

Notice how much the two examples look alike. Both give two numbers and ask for a single number back. The words together and next point at the lowest common multiple; the words greatest and same number in each point at the highest common factor.

Progress check

You have answered the chapter problem: the lighthouses next flash together after 36 seconds.

Now go back to what you wrote at the end of Section 3.1, where you decided which idea the question needed. Were you right? If you changed your mind, write one sentence saying which word in the question settled it — that word is what you look for next time.

You should now be able to

• find the highest common factor of two or more numbers from their prime factorisations;
• find the lowest common multiple the same way;
• say which of the two a word problem is asking for, and name the words that told you;
• check your answer against the size of the numbers you started with.