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.
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?
Are there any two-digit numbers which are the product of their digits?
The director of a bank has forgotten the combination to open the safe! He only remembers the first \(8\) out of \(10\) digits, and that the whole number was divisible by \(45\). Help him out and find all possible pairs of digits which could complete the combination. \[20242025**\]
Show that there are infinitely many composite numbers \(n\) such that \(3^{n-1}-2^{n-1}\) is divisible by \(n\).