Question

Suppose that GCD(a,b) = 1 and that a divides bc. Show that a must divide c.

Answer 1

Suppose a|c and b|c. Then c = am and c = bn.

Suppose that gcd(a,b) = 1. We may write it as a linear combination:

1 = ax + by

Observe that

1 = ax + by
c = acx + bcy
c = a(bn)x + b(am)y
c = ab(nx) + ab(my)
c = ab(nx + my).

Answer 2

hopes this helps

Answer 3

Why does c = am and c = bm?

Answer 4

And why did we suppose that a|c and b|c?