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 \(n\).
We have two rectangles: the first one has sides of length \(a\) and \(c\), and the second rectangle has sides of length \(b\) and \(d\).
Imagine that the difference in their side lengths, i.e: \(a-b\) and \(c-d\) are both divisible by \(11\). Show that the difference in their areas, i.e: \(ac-bd\), is also 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 \(y\).
Deep in a forest there is a small town of talking animals. Elephant, Crocodile, Rabbit, Monkey, Bear, Heron and Fox are friends. They each have a landline telephone and each two telephones are connected by a wire. How many wires were required?