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Number Theory

Greatest Common Divisor / Lowest Common Multiple

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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