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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 4

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 $$\gcd(k,2015)=1$$.

How many ordered pairs of positive integers $$(m, n)$$ satisfy

$\gcd (m^3, n^2) = 2^2 \cdot 3^2 \text{ and } \text{lcm}(m^2, n^3) = 2^4 \cdot 3^4 \cdot 5^6?$

This problem is shared by Muhammad A.

Details and assumptions

$$\gcd(a, b)$$ and $$\text{lcm}(a, b)$$ denote the greatest common divisor and least common multiple of $$a$$ and $$b$$, respectively.

Let $$m$$ and $$n$$ be positive integers such that 11 divides $$m+13n$$ and 13 divides $$m+11n$$.

What is the minimum value of $$m+n$$?

Consider the sequence $$50 + n^2$$ for positive integer $$n$$:

$51, 54, 59, 66, 75, \ldots$

If we take the greatest common divisor of 2 consecutive terms, we obtain

$3, 1, 1, 3, \ldots$

What is the sum of all distinct elements in the second series?

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