Find all solutions to the system of three congruence’s x=2 (mod? 3), x=1 (mod? 4

Question

Find all solutions to the system of three congruence’s x=2 (mod? 3), x=1 (mod? 4), x=3 (mod? 5). Hint: begin by finding the smallest x that satisfies all three conditions.

Explanation / Answer

Rewritting the problem we are looking for: x- 2 = 3*n ==> x=3*n+2 x-1 = 4*k == > x=4*k+1 x-3 = 5*r ==> x=5*r+3 So we are trying to solve this problem for integer values of n,k and r. You could just start at 1, and keep going - however we can eliminate some values with some detective work. Looking at 3*n+2, we see that this will give us both odd and even answers, so its no help. Looking at 4*k+1, we see this will give us ONLY odd answers. Thus X MUST be odd. Looking at 5*r+3, we see that if we choose "r" to be odd, will give us an even (an odd*odd+odd = even - try it for yourself), thus we must choose r to be even. So working from that you just start with even numbers - 2 is a good place. So as not to spoil the surprise - the number is between 2 and 22, for "r". So x is between 13 and 113.

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