Number Theory: Simple Divisibility & GCD

  • #1

Homework Statement

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

Homework Equations

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 [tex] d\not=1 [/tex]. 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?
  • #2
Yes, this is a valid argument.

And the converse is not true: consider 9.1+7.(-1)=2, but (9,7) is not 2...
  • #3
Thank you very much!

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