If an integer a is such that a2 is divisible by 3 then a^21 is divisible by 3. prove by direct method
1,690
Solution 1
$$a^21=(a1)(a+1)=(a1)(a2+3)=(a1)(a2)+3(a1)$$
So, $a^21$ will be divisible by $3$ if $a1$ or $a2$ is so
Solution 2
Hint $\ \ 3\mid \color{#c00}{a2}\ \Rightarrow\ 3\mid\!\! \overbrace{a+1}^{\large \ \color{#c00}{a2}\,+\,3}\!\!\mid a^21$
Solution 3
We're given
$$a2=3k\iff a= 2+3k\implies a^21=(a+1)(a1)=(3+3k)(a1)=3(1+k)(a1)\ldots$$
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Sudeep Acharya
Updated on August 09, 2022Comments

Sudeep Acharya about 1 year
How to prove that if a is number such that $a2$ is divisible by $3$ then $a^21$ is divisible by $3$ using direct method.
I know if $a = 2$ then $a2 = 0$ is divisible by $3$ and $2^21 = 3$ is divisible by $3$ but how to prove it using direct method.

Jlamprong almost 10 yearsHint: $a^21=(a2)(a+2)+3$


DonAntonio almost 10 yearsIf the OP knew even very basic modular arithmetic his question would be completely trivial...

mathlove almost 10 yearshmm, I agree with you, but I hope this might be the first step for the OP to learn it...

Sudeep Acharya almost 10 yearsyes i can do that thank you.

lab bhattacharjee almost 10 years@Downvoter, would you mind sharing the mistake.