need help with greatest common divisor, please.....( algebra help please)?

Question

prove that if gcd(ab,c)=1, then gcd(a,c)=gcd(b,c)=1


help please...
i really don't get how to prove this question..

thanks a lot...

Answer 1

Remember that the greatest common divisor of any two integers can be written as a linear combination of these integers with integer coefficients.
If gcd(ab,c) = 1, then there exist two integers s and t such that
1 = s(ab) + tc = (sb)a + tc
This implies that gcd(a,c) = 1

Also, since 1 = s(ab) + tc, then
1 = (sa)b + tc
This implies that gcd(b,c) = 1
QED

It looks harder than it really is.

Note that a, b and c do not have to be prime.
For example, let a=6, b=8 and c = 49
6, 8 and 49 are all composite (not prime)
ab = 6*8 = 48
gcd(48,49) = 1

Answer 2

If GCD of ab and c =1 then a,b, and c must all be different prime numbers. This is the only way that 1 can be the greatest common divisor.

Since all are prime, then
GCD of a and c must be 1 and GCD of b and c must be 1.

Let a = 3, b = 5 , and c =7 ( all different primes).

Then GCD ab and c = GCD of 15 and 7 = 1

GCD of a and c = GCD 3 and 7 = 1

GCD of b and c = GCD of 5 and 7 =1

It is all due to the fact that a,b,c are different prime numbers since their GCD is 1.

Answer 3

Assume gcd(a,c)=p. Then p divides c and ab. So p | gcd(ab,c). Hence p|1 and p=1. Similarly for gcd (b,c).

Answer 4

1=ab(k1)+c(k2) by Bezout's Lemma

d=gcd(a,c) divides a and divides c so it divides the RHS, hence d divides 1 and must be 1 since d is an integer.

Likewise, for gcd(b,c)

Answer 5

let A = A1*A2*..*Ai*1
let B = B1*B2..*Bj*1
let C = C1*C2..*Ck*1

if gcd(AB,C) = 1 it means that there are no
element in {A1 to Ai} and {B1 to Bj} equal to an element in {C1 to Ck} which mean that gcd(A,C) is 1 and gcd(B,C) is 1.