Divisibility problem 244

Algebra Level 2

Let a 2 + a + 1 a^2 + a + 1 be divisible by 3. If a a is divided by 3, what will be the remainder?

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Shefali Sharma by Shefali Sharma , 20, India

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1 solution

Chew-Seong Cheong
Sep 30, 2018

a 2 + a + 1 ≡ 0 (mod 3) Given a 2 + a ≡ − 1 (mod 3) a 2 + a ≡ 2 (mod 3) ⟹ a ≡ 1 (mod 3) \begin{aligned} a^2 + a + 1 & \equiv 0 \text{ (mod 3)} & \small \color{#3D99F6} \text{Given} \\ a^2 + a & \equiv -1 \text{ (mod 3)} \\ a^2 + a & \equiv 2 \text{ (mod 3)} \\ \implies a & \equiv \boxed 1 \text{ (mod 3)} \end{aligned}

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