Problems

Age
Difficulty
Found: 35

The numbers \(x,a,b\) are natural. Show that \(gcd(x^a -1,x^b-1) = x^{gcd(a,b)}-1\).

Let \(p\) be a prime number bigger than \(3\). Prove that \(p^2-1\) is a multiple of 24.

Is it possible to construct a 485 × 6 table with the integers from 1 to 2910 such that the sum of the 6 numbers in each row is constant, and the sum of the 485 numbers in each column is also constant?

Let \(p\) be a prime number greater than \(3\). Prove that \(p^2-1\) is divisible by \(12\).

You meet an alien, who you learn is thinking of a positive integer \(n\). They ask the following three questions.

“Am I the kind who could ask whether \(n\) is divisible by no primes other than \(2\) or \(3\)?"

“Am I the kind who could ask whether the sum of the divisors of \(n\) (including \(1\) and \(n\) themselves) is at least twice \(n\)?"

“Is \(n\) divisible by 3?"

Is this alien a Crick or a Goop?