If an integer a is such that a-2 is divisible by 3 then a^2-1 is divisible by 3. prove by direct method

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Solution 1

$$a^2-1=(a-1)(a+1)=(a-1)(a-2+3)=(a-1)(a-2)+3(a-1)$$

So, $a^2-1$ will be divisible by $3$ if $a-1$ or $a-2$ is so

Solution 2

Hint $\ \ 3\mid \color{#c00}{a-2}\ \Rightarrow\ 3\mid\!\! \overbrace{a+1}^{\large \ \color{#c00}{a-2}\,+\,3}\!\!\mid a^2-1$

Solution 3

We're given

$$a-2=3k\iff a= 2+3k\implies a^2-1=(a+1)(a-1)=(3+3k)(a-1)=3(1+k)(a-1)\ldots$$

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Sudeep Acharya
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Updated on August 09, 2022

Comments

  • Sudeep Acharya
    Sudeep Acharya about 1 year

    How to prove that if a is number such that $a-2$ is divisible by $3$ then $a^2-1$ is divisible by $3$ using direct method.

    I know if $a = 2$ then $a-2 = 0$ is divisible by $3$ and $2^2-1 = 3$ is divisible by $3$ but how to prove it using direct method.

    • Jlamprong
      Jlamprong almost 10 years
      Hint: $a^2-1=(a-2)(a+2)+3$
  • DonAntonio
    DonAntonio almost 10 years
    If the OP knew even very basic modular arithmetic his question would be completely trivial...
  • mathlove
    mathlove almost 10 years
    hmm, I agree with you, but I hope this might be the first step for the OP to learn it...
  • Sudeep Acharya
    Sudeep Acharya almost 10 years
    yes i can do that thank you.
  • lab bhattacharjee
    lab bhattacharjee almost 10 years
    @Downvoter, would you mind sharing the mistake.