How many positive integers less than 101 are multiples of either 5 or 7, but not both at once?

There are 20 positive multiples of 5 less than 101. There are 14 positive multiples of 7 less than 101. However, the least common multiple of 5 and 7 is 35, and there are 2 positive multiples of 35 less than 101. This means there are 20 - 2 = 18 multiples of 5 that aren't multiples of 7, and 14 - 2 = 12 multiples of 7 that aren't multiples of 5, for a total of 18 + 12 = 30.