The problem:

How many minimum number of integers must be selected from the set $\{1, 2, 3, 4,\dots,99, 100\}$ to ensure that there are at least two integers in the selection such that one divides the other.

My sloppy thinking: If we select $25$ primes, none of them of course will divide the other. If, however, we choose a $26^{th}$ number, it must divide some or the other. So the answer is 26.

The actual answer, however was 51. Hints?