There are 100 notes of two types: \(a\) and \(b\) pounds, and \(a \neq b \pmod {101}\). Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.
Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.
Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).
Valerie wrote the number 1 on the board, and then several more numbers. As soon as Valerie writes the next number, Mike calculates the median of the already available set of numbers and writes it in his notebook. At some point, in Mike’s notebook, the numbers: 1; 2; 3; 2.5; 3; 2.5; 2; 2; 2; 2.5 are written.
a) What is the fourth number written on the board?
b) What is the eighth number written on the board?
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.