Question

Show by proof that except for 2 and 5, every prime can be expressed as 10k +1, 10k+3, 10k+7, or 10k+9, where k is an integer.

Answer 1

Seems like you need induction

Answer 2

You could approach the converse?

In other words, assume that no primes can be expressed in terms of those equations.

Hmm... no, I think that would only provide proof that some primes exist.

Answer 3

it is just an idea.

we can show every integer by this form: 10k+i ,(i=0.1,2...8.9)
we can prove that 10k, 10k+2,10k+4 , 10k+6,10k+8 are not prime (even nums; except 2 ).
also 10k+5 is multiple of 5 and not prime (except 5). so if A={10k+i, for i=0,2,4,5,6,8}, a prime number belong to complement of A; or {10k+i | i=1,3,7,9}

(exception 2 , 5)