Show that for any integer n then gcd 10n 3 5n 2 1Solution

Show that for any integer n then gcd (10n + 3, 5n + 2) = 1

Solution

Assume gcd(10n+3,5n+2)=g ,g is not equal to 1 for some integer n

Hence, g| 10n+3,g|5n+2

So, let 10n+3=pg,5n+2=qg so that, gcd(p,q)=1

10n+3=pg=10n+4-1=2(5n+2)-1=2qg-1

2qg-pg=1

g(2q-p)=1

Hence, g|1 but g is an integer and we assumed g is not 1

Hence a contradiction

HEnce, gcd(10n+3,5n+2)=1 for any integer n