# Divisibility

• May 21st 2010, 06:51 AM
pollardrho06
Divisibility
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.
• May 21st 2010, 06:58 AM
Failure
Quote:

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

This is a consequence of $\displaystyle A\backslash B=A\backslash(A\cap B)$, and $\displaystyle |X\backslash Y|=|X|-|Y|$, if $\displaystyle Y\subseteq X$.
• May 21st 2010, 10:47 AM
pollardrho06
Quote:

Originally Posted by Failure
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...
• May 21st 2010, 11:36 AM
wonderboy1953
Hint
Look for cycles.
• May 21st 2010, 02:22 PM
Soroban
Hello, pollardrho06!

Quote:

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.

• May 21st 2010, 02:46 PM
pollardrho06
Quote:

Originally Posted by Soroban
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!!