TITLE: Find a example such $\frac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016$
QUESTION [9 upvotes]: Assume $x,y,z$ be postive integers,and Find  one  example $(x,y,z)$ such
$$\dfrac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016$$

REPLY [4 votes]: Such a thing will never happen. Why? Because 2016 is divisible by 7. Let's see what can be said about the power of 7 in the prime decomposition of this expression. Those of $x,\;y,\;z,\;x+y,\;y+z,\;z+x$ which are not divisible by 7 themselves, contribute nothing. Those which are, contribute (add or detract) a multiple of themselves, and hence a multiple of 7. But $2016=2^5\cdot3^2\cdot7^1$, and 1 is not a multiple of 7.
(The same reasoning could be applied to 2 or 3, of course.)