Question

Find the gcd of 6, 14, 21 and express it in the form 6r 14s 21t for some integers r, s, and t

Answer 1

r = 20; s = - 10; t = 1

Explanation:

The GCD of 6, 14, 21.

We have to know that gcd(a, b, c) = gcd(gcd(a, b), c)

Also, gcd(a, b) = au + bv, u & v are integers.

Now, gcd(6, 14, 21) = gcd(gcd(6, 14), 21)

= gcd(2, 21)

= 1

Again gcd(6, 14) = 2 & gcd(2, 21) = 1

⇒ 6(-2) + 14(1) = 2 & 2(-10) + 21(1) = 1

Now, gcd(6, 14, 21) = gcd(gcd(6, 14), 21)

= gcd(2, 21)

= 2(-10) + 21(1)

= {6(-2) + 14(1)}(-10) + 21(1)

= 6(20) + 14(-10) + 21(1)

Therefore, we get: r = 20; s = - 10; t = 1

Hence, the answer.