SOLUTION: This is a Number Theory proof: If N = abc + 1, prove that (N, a) = (N, b) = (N, c) = 1. (N, a) means "the greatest common divisor of N and a." I have started the proof lik

Algebra ->  Proofs -> SOLUTION: This is a Number Theory proof: If N = abc + 1, prove that (N, a) = (N, b) = (N, c) = 1. (N, a) means "the greatest common divisor of N and a." I have started the proof lik      Log On


   

Question 935274: This is a Number Theory proof:
If N = abc + 1, prove that (N, a) = (N, b) = (N, c) = 1.
(N, a) means "the greatest common divisor of N and a."
I have started the proof like this:
Let d = (N, a).
Since d = (N, a), then d | N and d | a. "d divides N" and "d divides a".


Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Then a%2Fd=x and N%2Fd=M are integers.
N%2Fd=M--->N=dM
a%2Fd=x--->a=dx
system%28a=dx%2CN=abc%2B1%29 ---> N=dxbc%2B1
system%28N=dM%2CN=dxbc%2B1%29 ---> dM=dxbc%2B1--->dM-dxbc=1--->d%28M-xbc%29=1
Since M , x, b, and c are all integers, so is M-xbc ,
and since the product of integers d and M-xbc is 1 ,
they must both be 1 :
M-xbc=1 and more importantly d=1 .

The way that is proven for a ,
it can be proven for b and c
(but in a raesonable world it should not be required),
because they all play the same role with different names.