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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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