Number 34 please Consider the following conjecture CONJECTUR

Consider the following conjecture. CONJECTURE: lf gcd(a. b) = d and d\\c, then gcd (a, b, c) = d Test the conjecture on three different choices for a, b and c. Either prove the conjecture true, or clearly explain how you know it is false.

Solution

Lemma 2. gcd(a, b) = d and d/cProof. Let d = gcd(a, b). Since
d = gcd(a, b), d/a and d/b, so by Lemma 1, i.e. d is a common
divisor of |a| and |b|. Now we prove by contradiction that d is the greatest
such divisor. Assume there is some c > d such that d/c. Then by
Lemma 1, c/a and c/b. So c is a common divisor of a and b strictly greater than
the GCD of a and b, which contradicts the de nition of the GCD. Therefore we
must have gcd(a, b) = gcd(d). Also, the de nition of gcd(x, y) is clearly
symmetric in x and y, so gcd(a,b,c) = d. Hence proved.