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.

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- May 21st 2010, 06:51 AMpollardrho06Divisibility
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 AMFailure
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 AMpollardrho06
- May 21st 2010, 11:36 AMwonderboy1953Hint
Look for cycles.

- May 21st 2010, 02:22 PMSoroban
Hello, pollardrho06!

Quote:

Find the number of positive integers not exceeding 1000

that are divisible by 3 butby 4.*not*

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 3by 4.*and*

. . There are: .$\displaystyle \left[\frac{1000}{12}\right] \:=\:83$ multiples of 3 whichdivisible by 4.*are*

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

- May 21st 2010, 02:46 PMpollardrho06