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Modular Arithmetic and Elementary Algebra

where z is an integer then gcd(a, b) = gcd(b, c). Indeed any divisor of a and b will divide c, and conversely any divisor of b and c will divide a. We can compute c by taking the remainder after dividing a by b, i.e. c is a mod b. (We will discuss the mod operation in greater details in the next section, but at this point, we only need the definition of c as the remainder of dividing a by b.) But c < b < a and thus we have made progress by reducing the numbers we have to compute their gcd of. And therefore, we can proceed and express b as: