# Proofs : GCDs and Primes

1. (i) Find the gcd (210, 48) using factorizations into primes

(ii)Find (1234, 5678)

2. Prove that there are no integers x, y, and z such that

x^2 + y^2 + z^2 = 999

keywords: greatest comon divisor

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#### Solution Preview

Problem #1

(i) We use factorizations.

210 = 2*3*5*7

48 = 2*2*2*2*3

Thus gcd(210,48)=2*3=6

(2) To find gcd(1234,5678), we can use the Eucliean algorithm.

5678 = 4*1234 + 742

1234 = 1*742 + 492

742 = 1*492 + ...

#### Solution Summary

Proofs involving GCDs and Primes are provided. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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