For 10 points, name this operation which finds the largest number that divides both inputs.
This operation can be defined as the sum of the Euler totient functions for each natural number that is a factor of both inputs. For two integers a and b, there exists a linear combination of a and b that is equal this operation applied to a and b according to Bezout's identity. If this operation applied to two integers yields one, then they are coprime to each other. A recursive algorithm to perform this operation involves dividing each remainder by the divisor, and was formulated by Euclid. The product of the least common multiple and this operation yields the product of the two inputs. For 10 points, name this operation which finds the largest number that divides both inputs.
This operation can be defined as the sum of the Euler totient functions for each natural number that is a factor of both inputs.
greatest common factor [or greatest common divisor; or GCF; or GCD]