Least common multiple by listing

When you reach for it

Rosa stands on a cliff between two lighthouses. The North light flashes every 4 seconds and the South light every 6 seconds, and both flash at the moment she starts her stopwatch. Counting from then, the North light flashes at these times, in seconds:

These are the multiples of 4. The South light flashes at the multiples of 6:

The two flash together whenever a time is on both lists. Those times, 12, 24, 36 and so on, are the common multiples of 4 and 6, and the first of them, 12 seconds, answers Rosa’s question.

The least common multiple of two whole numbers, neither of them 0, is the smallest positive number that is a multiple of both. It is written \text{LCM}(a, b), so \text{LCM}(4, 6) = 12. Three or more numbers have one too: the smallest positive number that is a multiple of all of them. The word “positive” matters, because 0 = 4 \times 0 is a multiple of every number, and without it every least common multiple would be 0.

Reach for it whenever things repeat on different cycles and you want the first time they line up again: lights, bells, rotas, or a pile of 6 cm tiles beside a pile of 8 cm tiles. One quick sign that a question wants it: the answer is built by repeating the given numbers, so it is at least as big as the larger of them.

The procedure

A multiple of a whole number b is b times a whole number. Here only the positive ones count: b \times 1, b \times 2, b \times 3 and so on, as in factors and multiples. Each list climbs in steps of b, so it is already in increasing order. Every step below rests on that.

Listing both

Write out the multiples of each number in order, and stop at the first number that appears on both lists. That number is a common multiple, because it is on both lists. It is the least one, because a smaller common multiple would have come earlier on both lists, and you would have stopped there instead.

The lists always meet. The product a \times b is on the list for a (it is a times the whole number b) and on the list for b (it is b times a, by the commutative property). So the search never runs past a \times b. It often stops sooner: 4 \times 6 = 24 is on both lists above, but 12 came first.

Listing only the larger number

You need not write the second list at all. A number is a multiple of a exactly when a divides it with nothing left over. So list the multiples of the larger number and test each one by dividing by the smaller: the first that divides exactly is the least common multiple. Starting from the larger number saves work, because its list climbs faster and reaches the answer in fewer entries. For 4 and 6: 6 \div 4 leaves a remainder of 2, and 12 \div 4 = 3 exactly, so the answer is 12.

This also shows that the least common multiple is never smaller than the larger number b, since the smallest positive multiple of b is b itself. When the smaller number divides the larger, that first entry already passes the test, so the least common multiple is the larger number: for example \text{LCM}(5, 15) = 15.

For three or more numbers, list the multiples of the largest and test each one against all the others.

Every time they line up

At 12 seconds Rosa’s two lights have both just flashed, exactly as at the start. From there each repeats its own pattern, so the next shared flash comes 12 seconds later, at 24, then 36, and so on. There is none in between: a shared flash at 12 + t seconds would mean the pattern, run from the start, gives a shared flash at t seconds, earlier than 12, which is impossible. So the lights line up exactly at the multiples of the least common multiple.

GCF or LCM?

The greatest common factor is the other number two numbers share, and word problems mix the two. The question to ask is whether things are being cut up or built up.

• Splitting given amounts into equal parts, as large as possible or as many as possible, asks for the greatest common factor. The answer divides each number, so it is no bigger than the smaller one.
• Repeating cycles, or stacking until two totals match, asks for the least common multiple. The answer is a multiple of each number, so it is no smaller than the larger one.

Worked examples

Boundaries

The method needs whole numbers other than 0. The only multiple of 0 is 0, which the definition rules out, so 0 has no least common multiple here.

Listing is slow when the numbers are large and share no factor but 1. For 23 and 29 the list of multiples of 29 runs to 23 entries, the last of them \text{LCM}(23, 29) = 667. Prime factorizations find it without any list.

In a “when do they line up” problem the least common multiple answers the question only when both cycles start together. If one light flashes at 1, 5, 9, 13, \dots seconds (every 4, starting at 1) and the other at 0, 6, 12, 18, \dots, the first list holds only odd numbers and the second only even ones, so the lights never flash together at all.

Where this goes next

Adding \displaystyle \frac{1}{4} and \displaystyle \frac{1}{6} needs a denominator that both 4 and 6 divide, and the smallest one is \text{LCM}(4, 6) = 12. That is the least common denominator, and it is what makes adding fractions with unlike denominators work with small numbers. For larger numbers, Finding them from prime factorizations finds the least common multiple from prime factorizations and shows how it is tied to the greatest common factor.

Exercises

Find the least common multiple

1. 4 and 10
2. 6 and 9
3. 5 and 8
4. 12 and 16
5. 14 and 21
6. 9 and 27
7. 10 and 25
8. 16 and 20

Three numbers

9. 2, 5 and 6
10. 3, 4 and 8
11. 6, 8 and 9

When do they line up?

In each, everything happens together at the start.

12. Bruno sees two lighthouses flash every 6 seconds and every 14 seconds. When do they next flash together?
13. Two beams flash every 9 seconds and every 12 seconds. Give the next three times they flash together.
14. A lighthouse flashes every 15 seconds and a bell buoy rings every 20 seconds. When do they next happen together, and how many times has the lighthouse flashed by then?
15. Nikolai watches lights that flash every 12 seconds and every 18 seconds. How many times do they flash together in the first 5 minutes, counting a shared flash at exactly 5 minutes but not the start?
16. Bruno cleans the North lamp every 6 days and Rosa cleans the South lamp every 10 days. Both clean today. In how many days do they next clean on the same day?
17. Three lighthouses flash every 4, 5 and 6 seconds. When do all three next flash together?

GCF or LCM?

Say which one each question needs, then answer it.

18. Sana stacks red tiles 6 cm thick in one pile and blue tiles 8 cm thick in another, until the piles are the same height. What is the smallest such height, and how many tiles of each color does it use?
19. Bruno has 21 red flags and 35 white flags. He makes identical bundles using every flag, as many bundles as possible. How many bundles, and what is in each?
20. Rosa hears a foghorn every 35 seconds and sees a beam every 10 seconds, together at the start. When are they next together?
21. Kai cuts strips 45 cm and 60 cm long into pieces of one equal length, a whole number of centimeters, with nothing left over, as long as possible. How long is each piece, and how many are there?

Working backwards

22. One light flashes every 4 seconds. Another flashes every p seconds, where p is a whole number greater than 4 and less than 12. Both flash at the start, and they next flash together after 20 seconds. Find every possible p.

Reasoning

23. Explain why the least common multiple of two numbers is never smaller than the larger of them, and say when it equals the larger one.
24. Explain why a \times b is always a common multiple of a and b. Give a pair of numbers for which it is not the least common multiple.
25. Thandiwe says: “The lights flash every 6 and every 9 seconds, so they flash together every 6 + 9 = 15 seconds.” Explain what is wrong, and give the correct time.