Divisibility

**pollardrho06** · May 21st 2010, 06:51 AM

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** · May 21st 2010, 06:58 AM

Originally Posted by 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.

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

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

**pollardrho06** · May 21st 2010, 10:47 AM

Originally Posted by Failure

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

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

I don't see it...

**wonderboy1953** · May 21st 2010, 11:36 AM

Look for cycles.

**Soroban** · May 21st 2010, 02:22 PM

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: . $\left[\frac{1000}{3}\right] \:=\:333$ numbers divisible by 3.

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

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

**pollardrho06** · May 21st 2010, 02:46 PM

Originally Posted by Soroban

Hello, pollardrho06!

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

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

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

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

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