Two numbers when divided by a divisor leave reminders of 248

Fun question.

Say the two numbers are x and y, and divisor is a.

x divided by a leaves a remainder of 248. This means that x = a*N + 248, where N is the integer result of the division.
y divided by a leaves a remainder of 372. This means that x = a*K + 372, where K is the integer result of the division.

x+y divided by a leaves a remainder of 68. This means that x = a*M + 68, where M is the integer result of the division.

From definitions above:

x+y = (a*N + 248) + (a*K + 372) = a*(N+K) + 620.

a*(N+K) + 620 = a*M + 68
552 = a*(M-N-K)

We know that M, N, and K are integers and that a must be at least 373 (to leave a 372 remainder). The only possible value for (M-N-K) is 1.

Therefore, a = 552. B.