Number Theory

Greatest Common Divisor / Lowest Common Multiple

Greatest Common Divisor / Lowest Common Multiple: Level 4 Challenges


For \(m>1\), it can be proven that the integer sequence \(f_m(n) = \gcd(n+m,mn+1)\) has a fundamental period \(T_m.\) In other words, \[\forall n \in \mathbb{N}, \space f_m(n+T_m) = f_m(n).\] Find an expression for \(T_m\) in terms of \(m,\) and then give your answer as \(T_{12}.\)

When 2017 is divided by a 2-digit number, what is the largest possible remainder?

Bonus: Generalize this problem.

Find the largest integer \(n\) such that \(n\) is divisible by all positive integers less than \(\sqrt[3]{n}\).

Find the sum of all integers \(k\) with \(1\leq k\leq 2015\) and \(\gcd(k,2015)=1\).

What is the greatest common factor of all integers of the form \(p^4 - 1\) where \(p\) is a prime number greater than 5?


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