solve $54 x + 16 y = 2400$ for integer values of x,y

Question

How to get integer values for x and y that satisfy: $$54 x + 16 y = 2400$$

Someone told me that I can do it using Euclid-Wallis algorithm, but I don't understand it so, if there isn't any else solution, can you explain it to me? I'm elementary.

Answer 1

First simplify it to $27x+8y=1200$.

Then $27x = 1200 - 8y = 8(150-y)$.

Since $27$ and $8$ are coprime, this implies that $x=8z$ and so $27z=150-y$.

Therefore, the general solution is $x=8z$, $y=150-27z$.

Answer 2

Its a line, so solve for y

y = 150 - 27/8x

which has a y intercept of 150 and an x intercept of ~44. So sample every integer between 0 and 44, and by trial and error you will get all integer roots.

But for this particular case, I can do you one better. The only x integers will have a factor of 8. So {0, 8, 16, 24, 32, 40}. In fact, with any line, the integer solutions will be found at values where both x and y are integers, which in our case will only happen if x is a factor of 8. Therefore there are 6 integer roots. (And to find y plug the x back into the original equation to solve in O(1))