Question

Find the number of positive integers not exceeding 108 that are not divisible by 5 or by 7.

Answer 1

75

Explanation:

The set of positive integer not exceeding 108 divisible by 5 is

`D_5=\{5 \quad 10 \quad 15 \quad 20 \quad 25 \quad 30\quad 35 \quad 40 \quad 45 \quad 50 \quad 55 \quad 60 \quad 65\\ \quad 70 \quad 75 \quad 80 \quad 85 \quad 90 \quad 95 \quad100 \quad 105\}`

and the set of positive integer not exceeding 108 divisible by 7 is

`D_7=\{7 \quad 14 \quad 21 \quad 28 \quad 35 \quad 42 \quad 49 \quad 56 \quad 63 \quad 70 \quad 77 \quad 84 \quad 91 \quad 98 \quad 105\}`

Moreover, there are exactly three positive numbers not exceedng 108 that are divisible by both 5 and 7, i.e,

`D_5 \cap D_7=\{37 \quad 70 \quad 105\}`.

Also note that the size of
`D_5` is
`\#D_5=21` , the size of
`D_7` is
`\#D_5=15` and
`\# D_7 \cap D_5 = 3`.

On the other hand, If a positive integer not exceding 108 is not divisible by 5 or 7, then it doesn't belong to any of this sets. Therefore, the number of positive interges not exceding 108 that are not divisible by 5 or 7 is equal to

`108 -(\#D_7 + \# D_5 - \# D_7 \cap D_5)=108 -(21+15-3)=75`