Again an equation with two variables!
x ( x + 2 ) ( x + 8 ) = 3 y
Find the sum of all non negative integers x and y such that the equation above is fulfilled.
The answer is 4.
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2 solutions
Moderator note:
Great observation about the "parity" of the terms!
Good one! I Did it exactly the same way
Let x = 3 u , x + 2 = 3 v , x + 8 = 3 t
So u + v + t = y
Then 3 v − 3 u = 2 and 3 t − 3 u = 8
It follows that 3 u ( 3 v − u − 1 ) = 2 and 3 u ( 3 t − u − 1 ) = 8
Hence u = 0 and 3 v − 1 = 2 , 3 t − 1 = 8
Therefore v = 1 , t = 2
The solution is x = 1 , y = 3
Moderator note:
Good approach with taking the difference to bound the values further.
There is a much simpler solution:
Note that x and x + 2 are not congruent mod 3 and so the product of two can never be a power of 3, except in the case where x = 1 ; and so it turns out that x + 2 = 3 then, and conveniently x + 8 = 9 , leading to the only solution ( x , y ) = ( 1 , 3 ) .