If d gcd a m and dc then show that the congruence a x c mo

If d = gcd (a, m) and d|c, then show that the congruence a x = c (mod m) is equivalent to ax/d = c/d (mod m/d).

Solution

Given that d = gcd of (a,m)

i.e. a= dn and m = do for some integers n and o

Since d divides c, c = dk

Or a = cn/k and m = co/k

Or a = cn/k

a/d = n   

m/d=0

Hence it follows that

ax /x and c/d are congruent under mod m/d