Question

How many integers between $1$ and $200$ are multiples of both $3$ and $5$ but not of either $4$ or $7$

Answer 1

There are only 13 multiples of both 3 and 5, since

$200 = 3\cdot5\cdot13 + 5$

From these integers, we eliminate any that are also divisible by 4 or 7.

$3\cdot5\cdot4 = 60 \implies \{60,120,180\}$

$3\cdot5\cdot7 = 105 \implies \{105\}$

so there are 13 - 4 = 9 such integers.