Prove $\forall a,b,c \in \mathbb Z$, if $ab+ac\equiv 3\pmod 6$ then $b \not\equiv c \pmod 6$ by contraposition
136367
Prove $ ∀a, b, c ∈ Z,$ if $ab + ac ≡ 3$ (mod 6) then b $\not\equiv$ c (mod 6).
4141
Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
112541
Prove that $n^2 + 2^n$ is composite if $n\not\equiv3\pmod{6}$.
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
88036
Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
61860
Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$.
145752
Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
141369
If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$. Demonstration.
104124
Prove or disprove: $A-(B-C) = (A-B)-C$
100426
Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
79056
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$
21726
Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
32253
How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?