math1081-assignment/q3/part2.md

Part 2

Show that for all integers n, if n is a multiple of 27, then n+5 and n^2+2 are coprime.

2 expressions (for example a and b) are co-prime if gcd(a,b)=1.

By the division algorithm, if a=bq+r, then gcd(a,b)=gcd(b,r).

Consider that n^2+2 can be expressed in the form n^2-25+27, which in turn can be factorised as (n+5)(n-5)+27 through application of the "difference of two squares" identity.

If n is a multiple of 27, n=27k, where k\in\mathbb{Z}. As a consequence, n+5=27k+5\space(1).

Using the division algorithm, gcd(n^2+2,n+5)=gcd(n+5,27)\space(2).

Given (1), equation (2) can be simplified to gcd(5,27). 5 is a prime number, and not a factor of 27, meaning the GCD must be 1.

Since the greatest common divisor of n^2+2 and n+5 is 1, the two expressions are co-prime.