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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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