150 Followers Question

Number Theory Level 3

How many positive integers less than or equal to 2015 are divisible by 3, but are neither divisible by 5 nor by 7?

461 512 506 455

1 solution

Arulx Z
Jan 20, 2016

Number divisible by 3 under 2015 are 671.

Numbers divisible by 5 in the list are also divisible by $3 \cdot 5 = 15$ . Numbers divisible by 15 under 2015 are 134.

Numbers divisible by 7 in the list are also divisible by $3 \cdot 7 = 21$ . Numbers divisible by 21 under 2015 are 95.

Total numbers divisible by 5 and 7 under 2015 are $134+95 = 229$ .

Numbers divisible by 7 and 5 are repeated twice. So removing multiples of both 7 and 5 once will remove the repeats. Numbers divisible by 7 and 5 are also divisible by $3 \cdot 5 \cdot 7 = 105$ . Number of multiples of 105 under 2015 are 19.

Therefore, numbers under 2015 which are multiple of 3, which are also multiple of 5 or 7 (or both) are $229 - 19 = 210$ .

Numbers under 2015 which are divisible by 3 and are not multiples of 5 and 7 are $671 - 210 = 461$

Moderator note:

Good approach accounting for these numbers using the Principle of Inclusion and Exclusion, where the universe is "numbers divisible by 3".

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