Greatest Common Divisor / Lowest Common Multiple: Level 2 Challenges Practice Problems Online

$\large n! + 1 \qquad \qquad (n+1)! + 1$

If $n$ is apositive integer, find thegreatest common divisorof the two numbers above.

Notation: $!$ denotes thefactorialnotation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

Find the sum of all natural numbers $n$ such that

$\text{lcm}(1,2,3,\ldots, n) = n!$

Notations:

$\text{lcm}(\cdot)$ denotes thelowest common multiplefunction.
$!$ is thefactorialnotation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

If $x$ and $y$ are integers, then which of the following expressions could be equal to 2018?

24x + 42y

28x + 82y

48x + 84y

$\begin{aligned} (\underbrace{7+7+7+\cdots+7}_{46 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{11 \text{ times}}) &= 3 \\\\ (\underbrace{7+7+7+\cdots+7}_{21 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{5 \text{ times}}) &= 2 \end{aligned}$

Is it also possible to find positive integers $m$ and $n$ such that $(\underbrace{7+7+7+\cdots+7}_{m \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{n \text{ times}}) = 1 ?$

Positive integersfrom 1 to 3141 (inclusive) are written on a blackboard. Two numbers from the board are chosen, erased, and theirgreatest common divisoris written.

This is repeated until only one number remains on the blackboard. What is the maximum possible value of this number?

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Greatest Common Divisor / Lowest Common Multiple: Level 2 Challenges