21!

Algebra Level 2

2 1 2 + 21 + 1 2 1 3 − 1 = P Q \frac{21^2 + 21 + 1}{21^3-1}= \frac PQ

The equation above holds true for coprime positive integers P P and Q Q . What is P + Q P+Q ?


The answer is 21.

Ethan Song by Ethan Song , 22, USA

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2 solutions

Zach Abueg
Jul 3, 2017

x 3 − 1 = ( x − 1 ) ( x 2 + x + 1 ) ⟹ x 2 + x + 1 x 3 − 1 = 1 x − 1 Let x = 21 ⟹ 2 1 2 + 21 + 1 2 1 3 − 1 = 1 20 \displaystyle \begin{aligned} x^3 - 1 & = \left(x - 1\right)\left(x^2 + x + 1\right) \\ \implies \frac{x^2 + x + 1}{x^3 - 1} & = \frac{1}{x - 1} & \small \color{#3D99F6} \text{Let} \ x = 21 \\ \implies \frac{21^2 + 21 + 1}{21^3 - 1} & = \boxed{\displaystyle \frac{1}{20}} \end{aligned}

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James Bacon
Sep 22, 2018

The answer is in the topic itself; so just give a try :D

Or else; just calculate:

2 1 2 + 21 + 1 2 1 3 − 1 = 463 9260 = 1 20 ⟹ p + q = 20 + 1 = 21 \dfrac{21^2 + 21 + 1}{21^3-1} = \dfrac{463}{9260} = \dfrac{1}{20} \implies p + q = 20 + 1 = \large \boxed{ 21 }

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