# Divisibility

#### pollardrho06

Hi all.

I'm trying to figure out the following problem:

Find the number of positive integers not exceeding 1000 that are divisible by 3 but not by 4.

Help will be appreciated. Looking for a simple/elementary proof.

Thanks.

#### Failure

Hi all.

I'm trying to figure out the following problem:

Find the number of positive integers not exceeding 1000 that are divisible by 3 but not by 4.

Help will be appreciated. Looking for a simple/elementary proof.

Thanks.
Hint:
$$\displaystyle |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3 \text{ but not by } 4\}|$$
$$\displaystyle = |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3\}|-|\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 12\}|$$

This is a consequence of $$\displaystyle A\backslash B=A\backslash(A\cap B)$$, and $$\displaystyle |X\backslash Y|=|X|-|Y|$$, if $$\displaystyle Y\subseteq X$$.

• chiph588@

#### pollardrho06

Hint:
$$\displaystyle |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3 \text{ but not by } 4\}|$$
$$\displaystyle = |\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 3\}|-|\{x\in \mathbb{Z}_+\mid x \text{ divisible by } 12\}|$$

This is a consequence of $$\displaystyle A\backslash B=A\backslash(A\cap B)$$, and $$\displaystyle |X\backslash Y|=|X|-|Y|$$, if $$\displaystyle Y\subseteq X$$.
I don't see it...

Hint

Look for cycles.

#### Soroban

MHF Hall of Honor
Hello, pollardrho06!

Find the number of positive integers not exceeding 1000
that are divisible by 3 but not by 4.

Every third number is divisible by 3.
. . There are: .$$\displaystyle \left[\frac{1000}{3}\right] \:=\:333$$ numbers divisible by 3.

But every twelfth number is divisible by 3 and by 4.
. . There are: .$$\displaystyle \left[\frac{1000}{12}\right] \:=\:83$$ multiples of 3 which are divisible by 4.

Therefore, there are: .$$\displaystyle 333 - 83 \:=\:250$$ such numbers.

• pollardrho06

#### pollardrho06

Hello, pollardrho06!

Every third number is divisible by 3.
. . There are: .$$\displaystyle \left[\frac{1000}{3}\right] \:=\:333$$ numbers divisible by 3.

But every twelfth number is divisible by 3 and by 4.
. . There are: .$$\displaystyle \left[\frac{1000}{12}\right] \:=\:83$$ multiples of 3 which are divisible by 4.

Therefore, there are: .$$\displaystyle 333 - 83 \:=\:250$$ such numbers.

Wow!! The greatest integer function!! Gr8!! Thanks!!