Prove: Let 𝑥 ∈ ℤ and a, b be distinct primes. Then a|𝑥 and b|𝑥 if and only if ab|𝑥.
140412
If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$
153313
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
24098
Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
34954
If $11..11$($n$ times 1) is prime, prove that $n$ is also prime
92737
primes whose difference is a multiple of $n$
144592
how to prove the existence and uniqueness of the prime factorization of any natural number.
65826
Find a prime number $p$ and an integer $b<p$ such that $p$ divides $b^{p−1}−1$.
27784
Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.
19700
Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$
144622
Proof: If no prime less than or equal to $\sqrt{n}$ divides $n$, then $n$ is a prime
64148
If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
113994
Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$.
6332
If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
127296
prove that a|b and b|a iff a=+-b
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
135750
Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$
111490
Proof that every number has at least one prime factor