Greatest Common Divisor / Lowest Common Multiple
\[ \large n! + 1 \qquad \qquad (n+1)! + 1 \]
If \(n\) is a positive integer, find the greatest common divisor of the two numbers above.
\[\]Notation: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).
by
Anuj Shikarkhane
Find the sum of all natural numbers \(n\) such that
\[ \text{lcm}(1,2,3,\ldots, n) = n!\]
Notations:
- \(\text{lcm}(\cdot) \) denotes the lowest common multiple function.
- \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).
by
Muhammad Rasel Parvej
If \(x\) and \(y\) are integers, then which of the following expressions could be equal to 2018?
by
Pi Han Goh
\[\begin{align} (\underbrace{7+7+7+\cdots+7}_{46 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{11 \text{ times}}) &= 3 \\\\ (\underbrace{7+7+7+\cdots+7}_{21 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{5 \text{ times}}) &= 2 \end{align}\]
Is it also possible to find positive integers \(m\) and \(n\) such that \[ (\underbrace{7+7+7+\cdots+7}_{m \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{n \text{ times}}) = 1 ? \]
by
Pi Han Goh
Positive integers from 1 to 3141 (inclusive) are written on a blackboard. Two numbers from the board are chosen, erased, and their greatest common divisor is written.
This is repeated until only one number remains on the blackboard. What is the maximum possible value of this number?
by
Pranshu Gaba