Problems

The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than their product. Determine these numbers.

The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.

Solve the equation $3x+5y=7$ in integers.

Determine all integer solutions of the equation $3x-12y=7$.

Determine all the integer solutions for the equation $21x+48y=6$.

Prove there are no integer solutions for the equation $4^k-4^l={10}^n$.

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.

Solve the equation $x+\frac{1}{(y+1/z)}=10/7$ in natural numbers.