Question

How many positive integers less than 1000 a) are divisible by 7? b) are divisible by 7 but not by 11?

Answer 1

Answer:

  130

Step-by-step explanation:

The number of integers meeting the criteria can be found by counting them using a counting formula.

Divisible by 7

Integers divisible by 7 will have the form (7n), where n is some positive integer. The number of them less than 1000 can be found from ...

  7n ≤ 1000

  n ≤ 142.857

There are 142 integers less than 1000 that are divisible by 7.

Divisible by 7 and 11

Similarly, integers divisible by 7 and 11 will be of the form (77n), for some positive integer n.

  77n ≤ 1000

  n ≤ 12.987

There are 12 integers less than 1000 that are divisible by both 11 and 7.

Divisible by 7, not 11

The number of integers less than 1000 that are divisible by 7, but not 11, will be the difference of these numbers.

  142 -12 = 130 integers divisible by 7, but not 11.