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Algebra Level 3

Given that x x and y y are positive integers satisfying { ( x + y ) 2 = 169 x 2 − y 2 = − 39 \begin{cases} (x + y)^{2} = 169 \\ x^{2} - y^{2} = -39 \end{cases} What is the remainder when x y x^{y} is divided by y x y^{x} ?


The answer is 30177.

Jansen Wu by Jansen Wu , 22, Indonesia

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1 solution

Aryan Gaikwad
Feb 24, 2015

( x + y ) 2 = 13 2 x + y = 13 x 2 − y 2 = − 39 ( x + y ) ( x − y ) = − 39 { (x+y) }^{ 2 }={ 13 }^{ 2 }\\ x+y=13\\ \\{ x }^{ 2 }-{ y }^{ 2 }=-39\\ (x+y)(x-y)=-39

On substituting,

13 ( x − y ) = − 39 x − y = − 3 13(x-y)=-39\\ x-y=-3

Now solve the linear equation in 2 variables

x + y = 13 x − y = − 3 2 y = 16 y = 8 x + 8 = 13 x = 5 5 8 M O D 8 5 = 30177 x+y=13\\ x-y=-3\\ \\ 2y=16\\ y=8\\ \\ x+8=13\\ x=5\\ \\ { 5 }^{ 8 }\quad MOD\quad { 8 }^{ 5 }=\quad 30177

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