When Isabel was born, her mother was turning 20 years old. If Isabel and her mother live for another 100 years, how many times will their ages be multiples of each other?
When Isabel is $a$ years old, her mother is $20+a$ years old. If $a$ is a divisor of $20+a$, then $(20+a) / a=(20 / a)+1$ is an integer, and consequently, $20 / a$ is also an integer. Therefore, $a$ is a divisor of 20, and thus $a$ can be 1, 2, 4, 5, 10, or 20. So, there are a total of 6 times when their ages are multiples of each other.

| Isabel | 1 | 2 | 4 | 5 | 10 | 20 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Mother | 21 | 22 | 24 | 25 | 30 | 40 |

Notice that, after Isabel turns 20, her mother's age will never again be a multiple of Isabel's age.