Our task is to find the smallest number that is both a multiple of 10 and a multiple of 12. But what is a multiple? A multiple of a number is just that number times some integer.

The multiples of 10 are 10, 20, 30, 40, etc. The multiples of 12 are 12, 24, 36, 48, etc.

One way to solve this problem is to start listing the set of multiples of 10 and the set of multiples of 12 and see if we can find a number that is in both sets. The smallest such number is the least common multiple.

Another more rigorous way -- a method that will always work no matter how big the numbers are -- is to factor both numbers into their constituent primes, as follows (highlighted numbers are split into the factors in parentheses on the next line):

10
(2 * 5)
2 * 5


12
(4 * 3)
(2 * 2) * 3
2 * 2 * 3


Once we have the sets of prime factors for both numbers, we simply compose a new set, the smallest possible, that includes all the numbers from both sets of factored primes, then multiply the numbers in the new set together. The new set is:

2 * 2 * 3 * 5 = 60

Notice that the first set of factored primes contains one 2 and one 5. The new set also contains at least one 2 and one 5. The second set of factored primes contains two 2s and one 3. The new set also contains at least two 2s, and one 3. There are no numbers in the new set other than the ones we need to accommodate both sets of factored primes, so it is the smallest possible set.