prove that there does not exist n belongs to N such that n congruent 2 (mod 4) and n congruent 4 (mod 8)

Question

prove that there does not exist n∈N such that n≡2 (mod 4) and n≡4 (mod 8). Please help this was on my last exam but I was very lost and my professor did not have enough time to go over it in class.

Answer 1

I am not certain how you want such a proof developed, but I can give you the general strategy.

 

Any number congruent to 2 (mod 4) is going to be congruent to either 2 (mod 8) or 6 (mod 8).

 

4 (mod 8) is congruent to 0 (mod 4).

 

2 (mod 4) is congruent to any number 2 + 4n for any integer n.
4 (mod 8) is congruent to any number 4 + 8m for any integer m.

 

There is no pair of integers (n,m) such that

2 + 4n is congruent to 4 + 8m.