# Approach to determining most likely integer factors of a noisy measurement?

I have a quantity which is estimated from a number of noisy measurements. I know that the real underlying value must be some integer multiple of two quantities, e.g. $M = I_1C_1 + I_2C_2$ where $C_1$ and $C_2$ are some known (from theory) values and $I_1,I_2$ are the unknown integer multiples.

I'm casting this as a least squares minimisation problem, minimising over $I_1,I_2$ but I'm unsure of what minimisation technique is appropriate here? Is there some standard approach to this type of problem? I can get extreme values for the unknown integers by doing $ceil(M/C_i)$, so could then check every possible integer combination within that range. There are typically many such combinations though, so this would not be very efficient.