(((63 * 63) + (x * 63 + 3 * 63) + (3 * x)) mod 63) = 63

Question

This is a problem I saw while browsing around, I have no idea what the answer is and I am quite intrigued to find out.

Answer 1

 

x = -63 works

 

N mod 63 = 63 is equivalent to  N mod 63 = 0

 

(((63 * 63) + (x * 63 + 3 * 63) + (3 * x)) mod 63) = 0

((3969 + (63x + 189) + 3x) mod 63 = 0

(66x + 4158) mod 63 = 0

 

So, 66x + 4158 = 0 will satisfy the equation, solving for x gives x=-63

 

There are other solutions as well.   

 

x = ±21n,   {n=0,1,2,3,… }   all satisfy the original equation