SOLUTION: show that the only common factor of any two consecutive positive odd integers is 1.

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Question 27106: show that the only common factor of any two consecutive positive odd integers is 1.
Answer by kev82(151) About Me  (Show Source):
You can put this solution on YOUR website!
Hi,
The fundamental theorem of arithmatic tells us that any posotive intrger can be factorised as a product of primes. Using this lets write the consecutive odd integers as and
Arbitrarily choose a prime factor of called and consider . (If is a common factor of and then will be an integer.)

For this to be an integer, must be an integer. This is only true if . But and are both odd, so can't have a factor of two, thus the only common factor is one.
Hope that helps,
Kev