Discrete mathematics, Proofs

[nick711]

Prove the following statement, then write down its converse. “For all x ∈ Z, if x ≡ −1 (mod 7) then x^3 ≡ −1 (mod 7)."
Would this statement be false? I'm kinda confused on how to solve this question. I can only figure out that x ≡ -1 (mod 7) when x=6 and hence x^3 ≡ −1 (mod 7) is also true.

[Guest]

If you know what "mod 7" means, this is just simple arithmetic. Yes, if x= 6 then x= -1 (mod 7) but that is not sufficient since there are infinitely many integers, x, such that x= -1 (mod 7).

Saying that "x= -1 (mod 7)" is the same as saying "x= -1+ 7n for some integer n. Then x^3= -1^3+ 7(3)n+ 7^2(-3)n^2+ 7^3n^3= -1+ 7(3n- 21n^3+ 49n^3)= -1 (mod 7).

The "converse" of the statement "if p then q" is "if q then p".

The converse of this statement would be "if x^3= -1 (mod 7) then x= -1 (mod 7)". To prove that (which, apparently, you were NOT asked to do), look at the general x= a+ 7n. Then x^3= a^3+ 7n(3a^2)+ 7^2n^2(3a)+ 7^3n^3= a^3+ 7(3a^2n+ 3(7)an^2+ 7^2n^3)= a^3 (mod 7)= -1 (mod 7). So a^3= -1 (mod 7) and x= -1+ 7n, x= -1 (mod 7).