How many numbers between 5 and 25 have both 2 and 3 as factors?

There are two methods for determine the number of integers that have both 2 and 3 as factors between 5 and 25. The first method involves writing down all the integers between 5 and 25 and then counting the number of factors of both 3 and 3. Here is the list of integers between 5 and 25 {6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}. We can see that only the numbers {6,12,18,24} have 2 and 3 as factors.  There are only four numbers between 5 and 25 that have both 2 and 3 as factors.

The other way to solve this problem is to realize that since 2 and 3 are relatively prime, the only numbers that will have 2 and 3 as factors are multiples of the lowest common multiple of 2 and 3. Six is the lowest common multiple of 2 and 3. The only multiples of 6 between 5 and 25 are {6,12,18,24}. There are 4 numbers that have 2 and 3 as factors between 5 and 25.

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