Find the value of

Algebra Level 3

K = ( 1 0 4 + 324 ) ( 2 2 4 + 324 ) ( 3 4 4 + 324 ) ( 4 6 4 + 324 ) ( 5 8 4 + 324 ) ( 4 4 + 324 ) ( 1 6 4 + 324 ) ( 2 8 4 + 324 ) ( 4 0 4 + 324 ) ( 5 2 4 + 324 ) K = \frac { (10^{ 4 }+324)(22^{ 4 }+324)(34^{ 4 }+324)(46^{ 4 }+324)(58^{ 4 }+324) }{ (4^{ 4 }+324)(16^{ 4 }+324)(28^{ 4 }+324)(40^{ 4 }+324)(52^{ 4 }+324)}


The answer is 373.

Juan Pablo Cabeza by Juan Pablo Cabeza , 27, Chile

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1 solution

Ruslan Abdulgani
Jan 20, 2015

We can write x^4+324) = (x2 + 6x + 18)(x2- 6x + 18) =(x(x+6) + 18)(x(x-6) + 18).

So ((10(16)+ 18)(10(4)+ 18)(22(28)+ 18)(22(16)+ 18)…(46(52) + 18)(58(64) + 18)(58(52) + 18))/((4(10)+ 18)(4(-2)+ 18)(16(22)+ 18)(16(10)+ 18)……(52(58) + 18)(52(46) + 18))

Most of the factors are cancelled each other except, ((58(64) + 18))/((4(-2)+ 18) ) = 373

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