Problems

Let \(n\) be an integer such that \(n^2\) is divisible by \(2\). Prove that \(n\) is divisible by \(2\).

Let \(n\) be an integer. Prove that if \(n^3\) is divisible by \(3\), then \(n\) is divisible by \(3\).

The numbers \(x\) and \(y\) satisfy \(x+3 = y+5\). Prove that \(x>y\).

The numbers \(x\) and \(y\) satisfy \(x+7 \geq y+8\). Prove that \(x>y\).

Can three points with integer coordinates be the vertices of an equilateral triangle?

Prove that there are infinitely many natural numbers \(\{1,2,3,4,...\}\).

Prove that there are infinitely many prime numbers \(\{2,3,5,7,11,13...\}\).

Is it possible to colour the cells of a \(3\times 3\) board red and yellow such that there are the same number of red cells and yellow cells?

Prove the divisibility rule for \(25\): a number is divisible by \(25\) if and only if the number made by the last two digits of the original number is divisible by \(25\);
Can you come up with a divisibility rule for \(125\)?

Which of the following numbers are divisible by \(11\) and which are not? \[121,\, 143,\, 286, 235, \, 473,\, 798, \, 693,\, 576, \,748\] Can you write down and prove a divisibility rule which helps to determine if a three digit number is divisible by \(11\)?