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Number Theory Level 3

Let x , y x, y and z z be positive integers less than 40 such that ( x / z ) 3 + ( y / z ) 3 = 6 (x/z)^3 + (y/z)^3 = 6

Find x + y + z x + y + z .


The answer is 75.

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
Feb 4, 2017

( 17 / 21 ) 3 + ( 37 / 21 ) 3 = 6 (17/21)^3 + (37/21)^3 = 6

So x = 17 x= 17 , y = 37 y = 37 and z = 21 z = 21

17 + 37 + 21 = 75 17 + 37 + 21 = 75

1 Helpful 0 Interesting 0 Brilliant 0 Confused

It's hard to see any simple way of solving this other than some educated trial and error. The next smallest suitable value for z z is 960540 , but I'm not sure if this involves positive x , y x,y , given the discussion here .

Brian Charlesworth - 4 years, 4 months ago

Is this the only solution?

Pi Han Goh - 4 years, 4 months ago

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There are an infinite number of solutions, but the next smallest is very large, as seen in the links I've provided in my comment.

Brian Charlesworth - 4 years, 4 months ago

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