The great triplets

Number Theory Level 3

Find the number of all triplets $(x, y, z)$ of positive integers satisfying

$2^{x} + 2^{y} + 2^{z} = 2336$

please post your innovative solutions too

1 solution

U Z
Oct 11, 2014

$2^{x} + 2^{y} + 2^{z} = 2^{5}.73$

$\frac{2^{x} + 2^{y} + 2^{z}}{2^{5}}$ = 73

RHS is odd

LHS is even

therefore any one of the

$2^{x - 5}$

$2^{y - 5}$

$2^{z - 5}$ is 1

let

$2^{x - 5}$ = 1 therefore x =5

$2^{y - 5} + 2^{z - 5} = 72$

$2^{y - 8} + 2^{z - 8} = 9$

once again let $2^{y - 8} = 1$ therefore y = 8

z = 11

thus (x , y , z) = (5, 8 ,11) , (8,5,11),(11, 5,8),(5,11,8),(8,11,5)(11,8,5)