abstract algebra help please?
Prove or give a counterexample: For all integers m and n, if there exist integers q and r such that n = qm + r and r|m, then r is the greatest common divisor of m and n. Disregard whether r is positive or negative; we consider the GCD to be defined only up to units.
Comments
Yes. Let d = gcd(m,n). Then d|m and d|n. Since r = n - qm, it follows that d|r. On the other hand, since r|m, r|(qm + r) and therefore r|n. Since r|m and r|n, r|d. Since d|r and r|d, r = +/- d.