How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

Question

Chetanya Joshi · Accepted Answer

To solve this, we use the principle of inclusion-exclusion.
Let A be the set of multiples of 5.Let B be the set of multiples of 4.Let C be the set of multiples of 7.
We need to find |A - (B union C)|.
The number of multiples of 5 less than 1000:|A| = 999/5 = 199
The number of multiples of 4 less than 1000:|B| = 999/4 = 249
The number of multiples of 7 less than 1000:|C| = 999/7 = 142
The number of multiples of 20 (LCM of 5 and 4) less than 1000:|A &cap; B| = 999/20 = 49
The number of multiples of 35 (LCM of 5 and 7) less than 1000:|A &cap; C| = 999/35 = 28
The number of multiples of 140 (LCM of 5, 4, and 7) less than 1000:|A &cap; B &cap; C| = 999/140 = 7
Using inclusion-exclusion, we get:|A - (B union C)| = |A| - |A &cap; B| - |A &cap; C| + |A &cap; B &cap; C|= 199 - 49 - 28 + 7= 199 - 70= 129
Thus, there are 129 positive integers less than 1000 that are multiples of 5 but NOT of 4 or 7.