Problems

Let $a$, $b$, $c$ be integers; where $a$ and $b$ are not equal to zero.

Prove that the equation $ax+by=c$ has integer solutions if and only if $c$ is divisible by $d=\mathrm{GCD}(a,b)$.

Numbers $a,b,c$ are integers with $a$ and $b$ being coprime. Let us assume that integers $x_0$ and $y_0$ are a solution for the equation $ax+by=c$.

Prove that every solution for this equation has the same form $x=x_0+kb$, $y=y_0-ka$, with $k$ being a random integer.

Could it be that a) $\sigma (n)>3n$; b) $\sigma (n)>100n$?

Prove that for a real positive $\alpha$ and a positive integer $d$, $\lfloor \alpha /d\rfloor =\lfloor \lfloor \alpha \rfloor /d\rfloor$ is always satisfied.

Let $m$ and $n$ be integers. Prove that $mn(m+n)$ is an even number.

Solve the equations in integers:

a) $3x^2+5y^2=345$;

b) $1+x+x^2+x^3=2^y$.

Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.

Prove that if $p$ is a prime number and $1\le k\le p-1$, then $(\genfrac{}{}{0ex}{}{p}{k})$ is divisible by $p$.

Prove that if $p$ is a prime number, then $(a+b{)}^p-a^p-b^p$ is divisible by $p$ for any integers $a$ and $b$.

The numbers $1,2,\dots ,99$ are written on 99 cards. Then the cards are shuffled and placed with the number facing down. On the blank side of the cards, the numbers $1,2,\dots ,99$ are once again written.

The sum of the two numbers on each card are calculated, and the product of these 99 summations is worked out. Prove that the end result will be an even number.