Interesting, let's have a go for fun:
4x - Ay = B
=> Ay = 4x - B
=> y = 4/A * x - B
There are exactly 45 integer values of x between 0 and 500 that provide integer values of y.
Thus, these are values such that x/A has EXACTLY 45 integer values for some 45 values of x. (Since A and 4 are relatively prime, 4 does not contribute to this aspect)
So, in short, A needs to be an integer that is:
1. Relatively prime to 4.
2. Produces exactly 45 multiples in the range 0 to 500 exclusive.
How many values of A would do this?
For this, we can see that we need a number A, where 45*A < 500 but 46*A > 500
=> A < 11.11 but A > 10.87
The only number that satisfies this is 11.
Hence C - 1 possible value
I'm not sure if this is correct, but anyway was fun
~Abhay
Believe those who are seeking the truth. Doubt those who find it. -- Andre Gide
