When n is divided by 10, the remainder is a. when n is divided by 13, the remainder is b. what is n modulo 130, in terms of a and b? (your answer should be in the form ra+sb, where r and s are replaced by nonnegative integers less than 130.)

means there are integers such that



Multiplying the first equation by 13 and the second by 10 gives



Adding the two equations gives




and since (they are coprime), it follows that



where is the modular multiplicative inverse of 23 (as opposed to 1/23) modulo 130. By the Euclidean algorithm we have











which means 17 is the inverse of 23 mod 130, so