2016 UNCO Math Contest II Problems/Problem 4

Problem

Number Sieve

How many positive integers less than 100 are divisible by exactly two of the numbers 2, 3, 4, 5, 6, 7, 8, 9? For example, 75 is such a number: it is divisible by 3 and by 5, but is not divisible by any of the others on the list. (If you show the integers you find, then you may be assigned partial credit if you have accurately found most of them, even if you do not find all of them.)

Solution

There are eighteen such numbers: