Divisibility

Feb 2010
36
2
New Jersey, USA
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.
 
Jul 2009
555
298
Zürich
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 \(\displaystyle Y\subseteq X\).
 
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Reactions: chiph588@
Feb 2010
36
2
New Jersey, USA
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...
 

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)
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.

 
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Feb 2010
36
2
New Jersey, USA
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!!