Number Theory

Greatest Common Divisor / Lowest Common Multiple

Greatest Common Divisor / Lowest Common Multiple: Level 4 Challenges

           

lcm(1,77)+lcm(2,77)+lcm(3,77)++lcm(77,77)=? \text{lcm}(1,77) + \text{lcm}(2,77) + \text{lcm}(3,77) + \cdots + \text{lcm}(77,77) = \, ?

Notation: lcm(a,b)\text{lcm}(a,b) denote the Lowest Common Multiple of aa and bb.

For m>1m>1, it can be proven that the integer sequence fm(n)=gcd(n+m,mn+1)f_m(n) = \gcd(n+m,mn+1) has a fundamental period Tm.T_m. In other words, nN, fm(n+Tm)=fm(n).\forall n \in \mathbb{N}, \space f_m(n+T_m) = f_m(n). Find an expression for TmT_m in terms of m,m, and then give your answer as T12.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 nn such that nn is divisible by all positive integers less than n3\sqrt[3]{n}.

Find the sum of all integers kk with 1k20151\leq k\leq 2015 and gcd(k,2015)=1\gcd(k,2015)=1.

What is the greatest common factor of all integers of the form p41p^4 - 1 where pp is a prime number greater than 5?

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