Prove that if $p_1\mid a$ and $p_2\mid a$ then $p_1p_2\mid a$
93686
In a UFD, lcm = product for coprimes: $\,(a,b)=1,\ a,b\mid c\Rightarrow ab\mid c$
10236
Show that $a^n \mid b^n$ implies $a \mid b$
145041
Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
88982
Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
76267
Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.
114667
$d\mid a,b \iff d\mid\gcd(a,b) \ $ [GCD Universal Property]
103360
Prove $4p-3$ is a square knowing that $n\mid p-1$ and $p\mid n^3-1$, $p$ prime
70308
Prove that if $p$ and $p+2$ are both prime integers greater than $3$, then $6$ is a factor of $p+1$.
140412
If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$
64453
Prove if $(a,p)=1$, then $\{a,2a,3a,...,pa\}$ is a complete residue system modulo $p$.
90916
Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
1712
$ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]
77164
If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
70620
From $n=2$ to $n$-ary Euclid lemma $\ p\mid a_1\cdots a_n\Rightarrow\ p\mid a_1\,$ or $\,\cdots\, p\mid a_n$
30583
If $a\mid b$ and $b\mid a$, $a,b\in R$, show that $\exists w \in R^\times$ with $b=wa$
65534
Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$