The simplest possible: 1 equation, 1 unknown

Algebra Level 2

Let p be an odd prime number, and k be a positive integer.

Given that p 2 + p 1 + p 0 + 1 / k p^{2} + p^{1} + p^{0} + 1/k = p 2 × p 1 × p 0 × 1 / k p^{2} \times p^{1} \times p^{0} \times 1/k , what is k?

k cannot be determined k = p k = p + 1 k = p - 1

Denton Young by Denton Young , 47, USA

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3 solutions

Md Omur Faruque
Jul 11, 2015

Suppose, a = 1 k \frac{1}{k} .

As p 0 = 1 p^{0}=1 , we can rewrite the equation as: p 2 + p + 1 + a = a p 3 p^{2}+p +1+a=ap^{3} p 2 + p + 1 = a ( p 3 − 1 ) p^{2} + p +1= a(p^{3}-1) a = p 2 + p + 1 p 3 − 1 a=\frac{p^{2} + p +1} {p^{3}-1} = p 2 + p + 1 ( p − 1 ) ( p 2 + p + 1 ) =\frac{p^{2} + p +1} {(p-1)(p^{2} + p +1)} = 1 p − 1 =\frac{1}{p-1}

So, k = 1 a = p − 1 k=\frac{1}{a}=\boxed{p-1}

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Kevin Tran
Nov 6, 2016

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Denton Young
Jul 10, 2015

p 2 + p 1 + p 0 + 1 / k p^{2} + p^{1} + p^{0} + 1/k = ( 1 / k ) ( k p 2 + k p + k + 1 ) (1/k) (kp^{2} + kp + k + 1)

p 2 × p 1 × p 0 × 1 / k p^{2} \times p^{1} \times p^{0} \times 1/k = p 3 / k p^{3}/k

so: p 3 p^{3} = k ( p 2 + p + 1 ) + 1 k(p^{2} + p + 1) + 1

p 3 − 1 p^{3} - 1 = k ( p 2 + p + 1 ) k(p^2 + p + 1)

k = p − 1 k = p-1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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