A robot is programmed to move along the number line starting at \(2\). At each second, the number by which it moves up by must be a factor of the number it’s currently on, but not \(1\). For example, if the robot gets to \(10\), then it can move forward by \(2\), \(5\) or \(10\) steps, going to \(12\), \(15\), or \(20\). What numbers can it land on, and what numbers can’t it land on?
A gang of three jewel thieves has stolen some gold coins and wants to divide them fairly. However, they each have one unusual rule: (i) The first thief wants the number of coins to be divisible by \(3\) so they can split it evenly. (ii) The second thief wants the number of coins to be divisible by \(5\) because she wants to split her share with her four siblings. (iii) The third thief wants the number of coins to be divisible by \(7\) since he wants to split his share amongst seven company stocks.
However, they’re stuck as the number of coins isn’t divisible by any of these numbers. In fact, the number of coins is \(1\) more than a multiple of \(3\), \(3\) more than a multiple of \(5\) and \(5\) more than a multiple of \(7\).
What’s the smallest number of coins they could have? (And if you’re feeling generous, how would you help them out?)
For which \(n\) is the expression \(n^4+4^n\) prime?
Find all solutions to \(x^2+2=y^3\) in the natural numbers.
McDonald’s used to sell Chicken McNuggets in boxes of 6, 9 or 20 in the UK before they introduced the Happy Meal. What is the largest number of Chicken McNuggets that could not be bought? For example, you wouldn’t have been able to buy 8 Chicken McNuggets, but you could have bought \(21 = 6+6+9\) Chicken McNuggets.
Show that the equation \(x^4+y^4=z^4\) cannot satisfied by integers \(x,y,z\) if none of them are 0.
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?
Prove that \(3\) always divides \(2^{2n}-1\), where \(n\) is a positive integer.