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 \(y\).
Catherine asked Jennifer to multiply a certain number by 4 and then add 15 to the result. However, Jennifer multiplied the number by 15 and then added 4 to the result, but the answer was still correct. What was the original number?
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?
Liz is 8 years older than Natasha. Two years ago Liz’s age was 3 times greater than Natasha’s. How old is Liz?
Solve the equation: \(|x-2005| + |2005-x|=2006\).
Between them, Jennifer and Alex shared the money they made from running a lemonade stand. Jennifer thought: “If I took \(40\%\) more money then Alex’s share would decrease by \(60\%\)”. How would Alex’s share of the profits change if Jennifer took \(50\%\) more money for herself?