1. The problem statement, all variables and given/known data 1. If a,b and c are natural numbers and a, b are coprime and a divides bc then prove that a divides c 2. Prove that the lcm of a,b is ab / gcd(a,b) 2. Relevant equations if a is a divisor of b then a = mb for a natural number m if a prime p is a divisor of ab then p is a divisor of a or a divisor of b 3. The attempt at a solution 1.since a is a divisor of bc so am = bc (m is a natural number) so a = (c)(b/m) so a/b = c/m Ok since a,b are coprime so a/b = a number that is not natural since a/b = c/m so c/m = a number that is not natural so c,m are coprime back to a = (c)(b/m) since a = (c)(b/m) which is a natural number, so bc must be a multiplie of m since c isn't a multiplie of m, b must be so so b is coprime with m now a/c = b/m since b is coprime with m a is coprime with c Q.E.D (Wanna check if my approach is correct or not) 2.Prove that the lcm of a,b is ab/gcd(a,b) let a = xm , b = ym (m = gcd(a,b)) ab/gcd(a,b) = xmym/m = xmy It is divisible by a and b so it satisfies being a multiplie here I gave up.