Question 27106
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 *[tex a=p_1p_2p_3\ldots] and *[tex b=2+a=2+p_1p_2p_3\ldots]

Arbitrarily choose a prime factor of *[tex a] called *[tex p_j] and consider *[tex b/p_j]. (If *[tex p_j] is a common factor of *[tex a] and *[tex b] then *[tex b/p_j] will be an integer.)

*[tex \frac{b}{p_j}=\frac{2}{p_j}+p_1p_2\ldots p_{j-1}p_{j+1}\ldots]

For this to be an integer, *[tex 2/p_j] must be an integer. This is only true if *[tex p_j=1,2]. But *[tex a] and *[tex b] are both odd, so can't have a factor of two, thus the only common factor is one.

Hope that helps,

Kev