Problems

Show that a perfect square can only have remainders 0 or 1 when divided by 4.

Convert 2000 seconds into minutes and seconds.

What is a remainder of $7780\times 7781\times 7782\times 7783$ when divided by $7$?

Tim had more hazelnuts than Tom. If Tim gave Tom as many hazelnuts as Tom already had, then Tim and Tom would have the same number of hazelnuts. Instead, Tim gave Tom only a few hazelnuts (at most five) and divided his remaining hazelnuts equally between $3$ squirrels. How many hazelnuts did Tim give to Tom?

Prove that $n^3-n$ is divisible by $24$ for any odd $n$.

Show that if numbers $a-b$ and $c-d$ are divisible by $11$, then $ac-bd$ and $ad-bc$ are also both divisible by $11$.

For how many pairs of numbers $x$ and $y$ between $1$ and $100$ is the expression $x^2+y^2$ divisible by $7$?

Seven robbers are dividing a bag of coins of various denominations. It turned out that the sum could not be divided equally between them, but if any coin is set aside, the rest could be divided so that every robber would get an equal part. Prove that the bag cannot contain $100$ coins.

Show that the equation $x^2+6x-1=y^2$ has no solutions in integer $x$ and $y$.

Solve the equation $\lfloor x^3\rfloor +\lfloor x^2\rfloor +\lfloor x\rfloor =\{x\}-1$.