Prove parts 13 where k m n q and r designate integers Let n

Prove parts 1-3, where k, m, n, q, and r designate integers. Let n > 0 and k > 0. If q is the quotient and r is the remainder when m is divided by n, then q is the quotient and kr is the remainder when km is divided by kn.

Solution

Given that when m is divided by n, if q is the quotient and r is the remainder . Then by division algorithm we have

m = nq + r where 0 < r < q

Let k>0. Let us multiply above equation by k we get,

km = (kn)q + kr and 0 < kr < kq (since k>0)

Therefore by division algorithm,

we can conclude that when km is divided by kn then q is the quotient and r is the remainder