Question

Prove that for any positive integer n, 6 evenly divides 7n - 1.

Answer 1

For any positive integer, n, 6 DOES NOT EVENLY divide 7n-1; 6 doesn't even divide the expression in all positive integers. For postive integers we have, n = 1,2, 3, .... Suppose n =1 then we have 7(1) -1 = 6/6 = 1 It divides it but not evenly since 1 is not an even number. Take n= 2; we have 7(2)-1 = 13/6 and take n =3; we have 7(3)-1 = 20/6 so it is really obvious that 6 doesn't divide all the expression

Answer 2

This is not true. Check $n=2$, which gives

$a_n=7n-1\implies a_2=13$

but 13 is prime, so in general $6\nmid a_n$.