BdMO National Junior 2008/4

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
samiul_samin
Posts:1007
Joined:Sat Dec 09, 2017 1:32 pm
BdMO National Junior 2008/4

Unread post by samiul_samin » Sun Feb 24, 2019 9:08 am

$p$ is a prime number and given that $p>3$.What be the remainder if $p^2$ divided by $12$?

samiul_samin
Posts:1007
Joined:Sat Dec 09, 2017 1:32 pm

Re: BdMO National Junior 2008/4

Unread post by samiul_samin » Sun Feb 24, 2019 1:15 pm

Answer:$\fbox 1$
Solution:
$p$ is not divided by $3$
$p$ is not divided by $4$
So,
\[p\equiv 1(mod 3)\]
\[p^2\equiv 1^2(mod 3)\]
\[p^2\equiv 1(mod 3)\]
Again
\[p\equiv 1(mod 4)\]
\[p^2\equiv 1^2(mod 3)\]
\[p^2\equiv 1(mod 4)\]
So,
\[p^2\equiv 1(mod 3\times4)\]
\[p^2\equiv 1(mod 12)\]

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