having problem in congruence

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tushar7
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Joined:Tue Dec 07, 2010 3:23 pm
having problem in congruence

Unread post by tushar7 » Sun Dec 26, 2010 2:17 pm

if $m= k_1\cdot k_2$
then $a\equiv b\bmod (k_1)$
$a\equiv b_1\bmod (k_2)$
the can i write that ..
$a\equiv b\cdot b_1\bmod (m)$
i am confused ??!

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Zzzz
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Re: having problem in congruence

Unread post by Zzzz » Sun Dec 26, 2010 3:02 pm

Its not true. For example, take $a=8, k_1=2, k_2=3$
Every logical solution to a problem has its own beauty.
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tushar7
Posts:101
Joined:Tue Dec 07, 2010 3:23 pm

Re: having problem in congruence

Unread post by tushar7 » Sun Dec 26, 2010 3:46 pm

is there any property of congruence close to this ?!

HandaramTheGreat
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Re: having problem in congruence

Unread post by HandaramTheGreat » Sun Dec 26, 2010 3:58 pm

two properties are like these...
$1.$ if $m=k_1\cdot k_2$ and $gcd\left(k_1, k_2\right)=1$...
$a\equiv b\bmod(k_1)$
$a\equiv b\bmod(k_2)$
then $a\equiv b\bmod(m)$

$2.$ $a\equiv b\bmod(m)$
$c\equiv d\bmod(m)$
then $ac\equiv bd\bmod(m)$
Last edited by HandaramTheGreat on Sun Dec 26, 2010 4:14 pm, edited 1 time in total.

tushar7
Posts:101
Joined:Tue Dec 07, 2010 3:23 pm

Re: having problem in congruence

Unread post by tushar7 » Sun Dec 26, 2010 4:03 pm

$ m $ doesnt have to be the lcm of $k_1$ and $k_2 $

HandaramTheGreat
Posts:135
Joined:Thu Dec 09, 2010 12:10 pm

Re: having problem in congruence

Unread post by HandaramTheGreat » Sun Dec 26, 2010 4:13 pm

HandaramTheGreat wrote:if $m=k_1\cdot k_2$ and $gcd\left(k_1, k_2\right)=1$...
this is only for first property...

tushar7
Posts:101
Joined:Tue Dec 07, 2010 3:23 pm

Re: having problem in congruence

Unread post by tushar7 » Mon Dec 27, 2010 1:31 pm

HandaramTheGreat wrote:
HandaramTheGreat wrote:if $m=k_1\cdot k_2$ and $gcd\left(k_1, k_2\right)=1$...
i know it was for the 1st property . so $m$ has to be the lcm of $k_1$ and $k_2$

HandaramTheGreat
Posts:135
Joined:Thu Dec 09, 2010 12:10 pm

Re: having problem in congruence

Unread post by HandaramTheGreat » Mon Dec 27, 2010 1:42 pm

m doesnt have to be the lcm of k1 and k2
then what's it? can't understand what you want to say...

tushar7
Posts:101
Joined:Tue Dec 07, 2010 3:23 pm

Re: having problem in congruence

Unread post by tushar7 » Tue Dec 28, 2010 12:58 pm

tushar7 wrote:
HandaramTheGreat wrote:
HandaramTheGreat wrote:if $m=k_1\cdot k_2$ and $gcd\left(k_1, k_2\right)=1$...
i know it was for the 1st property . so $m$ has to be the lcm of $k_1$ and $k_2$
this comment is right .....?

HandaramTheGreat
Posts:135
Joined:Thu Dec 09, 2010 12:10 pm

Re: having problem in congruence

Unread post by HandaramTheGreat » Tue Dec 28, 2010 3:31 pm

hm... if $k_1$ and $k_2$ is coprime then lcm of $k_1$ and $k_2$ is their product, isn't it?

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