A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by \(m\) squares to the right or by \(n\) squares to the left. For which \(m\) and \(n\) can it move to the next cell to the right?
Prove that the equation \(\frac {x}{y} + \frac {y}{z} + \frac {z}{x} = 1\) is unsolvable using positive integers.
In a room, there are three-legged stools and four-legged chairs. When people sat down on all of these seats, there were 39 legs (human and stool/chair legs) in the room. How many stools are there in the room?
A regular polygon has integer side lengths and its perimeter is 60. How many sides can it have?
Find positive integers \(x,y,z\) such that \(28x+30y+31z = 365\).
Given a piece of paper, we are allowed to cut it into 8 or 12 pieces. Can we get exactly 60 pieces of paper starting with a single piece?
Determine all integer solutions to the equation \(24a + 16b = 6\).
John’s father is 28 years older than John and next year he will be exactly three times the age of John. How old is John’s father?
You have an hourglass that measures 8 minutes and an hourglass that measures 12 minutes. How can you measure exactly 44 minutes with them?
Joe has two kinds of weights: 15 grams and 50 grams. He has an infinite supply of each type. Can you help him find a combination that is exactly 310 grams?