# Number Theory: Simple Divisibility & GCD

## Homework Statement

Prove that if N=abc+1, then (N,a)=(N,b)=(N,c)=1.

## The Attempt at a Solution

Assume N=abc+1. We must prove (N,a)=(N,b)=(N,c)=1. Proceeding by contradiction, suppose (N,a)=(N,b)=(N,c)=d such that $$d\not=1$$. Then we know, d | N and d | abc. Thus, from our assumption, we see that d | 1, a contradiction.

Is this a valid argument? Also, what is another way to prove this without using contradiction? Thanks, this is my first class with proofs. Also, I know "if (a,b)=d, then ax+by=d." Is the converse true?

## Answers and Replies

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Yes, this is a valid argument.

And the converse is not true: consider 9.1+7.(-1)=2, but (9,7) is not 2...

Thank you very much!