Let $a, b, c, d$ be positive integers satisfying $ab=cd$. Prove that $a+b+c+d$ ݀ is composite.
91285
Prove $a+b+c+d $ is composite
18772
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
55441
Check: For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$
11168
Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
97750
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
59479
For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
23066
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
128820
Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
55673
Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
63535
Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
76049
Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
149269
If $a,b,c,d,e,f$ are non negative real numbers such that $a+b+c+d+e+f=1$, then find maximum value of $ab+bc+cd+de+ef$
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
90670
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
45356
Are there any positive integers $a, b, c, d$ such that both $(a, b, c)$ and $(b, c, d)$ are Pythagorean triples?