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Number Theory

# Greatest Common Divisor / Lowest Common Multiple: Level 4 Challenges

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

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

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

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

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

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?

Let $$0 < a_1 < a_2 < \ldots < a_{100} \leq 200$$ be a set of 100 integers, such that the least common multiple of any two of them is strictly greater than 200. What is the smallest possible value of $$a_1$$?

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