BdMO National Junior 2008/4
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- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
$p$ is a prime number and given that $p>3$.What be the remainder if $p^2$ divided by $12$?
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- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
Re: BdMO National Junior 2008/4
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,
\[p^2\equiv 1(mod 3\times4)\]
\[p^2\equiv 1(mod 12)\]
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,
\[p^2\equiv 1(mod 3\times4)\]
\[p^2\equiv 1(mod 12)\]