Prove that if gcdb c 1 then gcda bc gcda bgcda cSolutionGC

Prove that if gcd(b, c) = 1, then gcd(a, bc) = gcd(a, b)gcd(a, c)

Solution

GCD(a,b)= 1

so 1 = ax + by

so c = acx + bcy ...1

now GCD(a,c) = 1

so 1 = ma+ nc = ma + n(acx+ bcy) ftom 1
= a ( m + ncx) + bc(ny)

as we can put 1 as linear combination of a and bc so Gcd(a,bc) = 1