Show that if $A ⊆ C$ and $B ⊆ D$, then $A × B ⊆ C × D$
41099
If $A \subseteq C$ and $B \subseteq D$ then $A \times B \subseteq C \times D$
11168
Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
52565
Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
120292
How to formally prove: if $d\mid da+b$, then $d\mid b$?
137626
If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
5149
Show $\binom{a}{b}\binom{b}{c}=\binom{a}{a-c}\binom{a-c}{a-b}$
5789
$ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$
20003
Prove for positive integers a,b,c and d (where b does not equal d), if gcd(a,b) = gcd(c,d) = 1, then a/b + c/d is not an integer
71299
Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
104124
Prove or disprove: $A-(B-C) = (A-B)-C$
152693
If $C|a$ and $C|b$ then $C|(ax+by)$
56399
Proof for $A \cup B = B$ if and only if $A \subset B$
101159
Prove If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n)
29935
If $a, b, c, d>0$ and $abcd=1$ prove that an inequality holds true
70307
$(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$
106416
If $n,m \in \mathbb{N}$ then there are $c,d$ such that $cd = (m,n)$, $(c,d) = 1$ and $(m/c,n/d) = 1$.
18119
Show that if a,b,c,d are positive then $\frac{ab}{a+b}+\frac{cd}{c+d}\le \frac{(a+c)(b+d)}{a+b+c+d}$