Throughout this problem a1 a2 an will be arbitrary integ

Throughout this problem, a1, a2, . . . , an will be arbitrary integers. As a generalization of gcd(m, n) and gcd(m, n, l), provide a definition for the Greatest Common Divisor for n elements, gcd (a1, a2, . . . , an). Then prove or disprove the following statements:

(b) god (a 1, a2, n) gca (a1, gcd (a2, a3, an

Solution

The GCD of n numbers represents the largest positive number which divides all the n numbers by leaving the remainder zero

b) Assume for all integers, we can write

a1 = k*p1, a2 = k*p2,...,an = k * pn

gcd(a1,a2,...,an) = k

gcd(a1,gcd(a2,a3,...,an)) = gcd(a1,k) = gcd(k*p1,k) = k

Hence both the statements are same