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Greatest Common Divisor / Lowest Common Multiple

What is the largest number that can divide two numbers without a remainder? What is the smallest number that is divisible by two numbers without a remainder? See more

Level 5

Let \(a\) and \(b\) be two distinct, natural numbers such that their harmonic mean, geometric mean and arithmetic mean are all natural numbers. Find the minimum value of the arithmetic mean.

How many pairs \((x,y)\) of positive integers with \(x \leq y\) satisfy \(\gcd(x,y) = 5!\) and \(\text{lcm}(x, y) = 50!\)?

\[ \large a_n = a_{n-1} + \gcd(n,a_{n-1}) \]

Consider the recurrence relation above with for \(n\geq2\) with \(a_1 = 7\). And define \(b_n= a_{n+1} - a_n \), find the number of composite numbers \(b_n\) for \(n\leq10^9 \).

For the sake of this question, take 1 as neither prime nor composite.

Find the largest integer \(n\) such that \(n\) is divisible by all positive integers less than \(\sqrt[4]{n}\).

\[\large \sum_{n=1}^\infty \dfrac{\gcd(n,2016)}{n^2}= \dfrac{a}{b}\pi^2\]

If the equation above holds true for positive integers \(a\) and \(b\), find \(a+b\).

Clarification:
\(\gcd(m,n) \) denotes the greatest common divisor of \(m\) and \(n\).

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