please answer in detail Prove that two consecutive integers

please answer in detail...

Prove that two consecutive integers don\'t have common factors greater than 1. That is, prove that

Solution

We need to prove:

                    two consecutive integers don, have common factors greater then one.

Let d=gcd(a,a+1)>1

   if supose take two consecutive numbers 100,101

gcd(100,101)=1

   d=1

if we take for k:

   d=gcd(k,k+1)=1 is true

if we take for n:

   d=gcd(n,n+1)=1 is true

by mathametical induction

                               for n+1: d= gcd(n+1,n+2)=1

       d always same so it\'s proved.