You may already know what a GCD and an LCM are, but let’s briefly review.

• What is the greatest common divisor of 12 and 26?

To do this, we need to take the prime factorization of both. The prime factorization of 12 is 2 x 2 x 3, and the prime factorization of 26 is 2 x 13. So, the only number they have in common is 2. 2 is the GCD.

• What is the least common multiple of 12 and 26?

Using the prime factorizations from before, we see that we need to add a 3 and another 2 to 26 to contain all of the prime factors of 12. If a number contains all of the prime factors of both numbers, it is a multiple. We are adding as few prime factors as possible to ensure that it is the least common multiple.

So, 2 x 3 x 26 = 156, which is the LCM of 12 and 26.

Now, let’s try some more problems.

• The GCD of two numbers is 5, and their LCM is 60. One number is 15. What is the other number?

To solve this problem, you can draw a Venn diagram. The GCD is in the middle, and the unique prime factors of each number are on the outsides. Each number is equal to the product of the numbers in its circle, which is the GCD, and the numbers in its unique section.

The product of all the numbers in the Venn diagram is equal to the LCM, which is 60. So, the number in the empty section of the Venn diagram has to be 60 / 15, which is 4. Taking the product of 4 and the GCD, which is 5, we get 20 as the value of the other number.

This Venn diagram is an extremely useful tool when solving problems related to the GCD and LCM.

• Three lights blink at different intervals. One blinks every 6 seconds, one every 9 seconds, and one every 15 seconds. They all blink together now. How many seconds will pass before they all blink at the same time again?

This problem is a little more complicated. To solve this problem, you have to realize that the next time they blink together will be the LCM of all three numbers. So, let’s start with taking the prime factorizations:

6: 2 x 3, 9: 3 x 3, 15: 5 x 3.

To get the LCM, we need to multiply 2 and an extra 3 to 15 so that all factors of all 3 numbers are accounted for.

2 x 3 x 15 = 90, so the lights will blink together in 90 seconds.