Number Theory

Greatest Common Divisor / Lowest Common Multiple

Greatest Common Divisor / Lowest Common Multiple: Level 2 Challenges

         

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

If nn is a positive integer, find the greatest common divisor of the two numbers above.

Notation: !! denotes the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

Find the sum of all natural numbers nn such that

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

Notations:

  • lcm()\text{lcm}(\cdot) denotes the lowest common multiple function.
  • !! is the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

If xx and yy are integers, then which of the following expressions could be equal to 2018?

(7+7+7++746 times)(29+29+29++2911 times)=3(7+7+7++721 times)(29+29+29++295 times)=2\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 mm and nn such that (7+7+7++7m times)(29+29+29++29n times)=1? (\underbrace{7+7+7+\cdots+7}_{m \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{n \text{ times}}) = 1 ?

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?

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