1
vote
1answer
34 views

Proof that 6 divides $a \in \mathbb{Z}, a(a^2 - 7)$

I am trying to prove a question from my tutorial sheet, is this an acceptable proof? Six cases exist: $$a,k \in \mathbb{Z}, a(a^2 - 7) = 6k \\\text{Proof:}\\ a = 0 \mod 6 \longrightarrow a^2 = 0 \mod ...
2
votes
2answers
53 views

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$

Last decimal digit of any perfect square must be $0,1,4,5,6$ or $9$ My Proof: Ten cases exist, yielding the following equalities: $$(1\mod{10})^2 = 1\mod{10}$$ $$(2\mod{10})^2 = 4\mod{10}$$ ...
0
votes
1answer
65 views

A nice group isomorphism

Show that $$k(\mathbb{Z}/n\mathbb{Z})\cong (\gcd(n,k)\mathbb{Z})/n\mathbb{Z}.$$ I want to see as many as possible proofs of this nice fact.
3
votes
3answers
148 views

If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$.

In this (btw, nice) answer to Twin primes of form $2^n+3$ and $2^n+5$, it was said that: If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$? I'm not familiar with these kind of calculations, so I'd like ...
3
votes
1answer
296 views

Is this a good proof of Wilson's theorem? — ($(n-1)!+1 \equiv_n 0$ iff n is prime)

Theorem: $(n - 1)! + 1 \equiv_n 0$ if and only if $n$ is prime. To prove that if $n$ is not prime this is not true is trivial, so I'm just interested in proving that this is true for all p: ...