Number Theory
# Greatest Common Divisor / Lowest Common Multiple

A teacher has 30 pens and 45 pencils that he wants to give to his students. If each student were to receive the same number of pens and the same numbers of pencils with none left over, then what is the maximum number of students that this teacher could have?

Miranda is an avid collector of stamps. She is trying to arrange her collection of stamps into neat rows. She found that when she arranged them in row of 2, 3, 4, 5, 6 or 7, she always came up 1 short.

What is the minimum number of stamps that Miranda has?

What positive integer, $x,$ satisfies these equations?

$\text{gcd}(391,x) = 23$

$\text{lcm}(391,x) = 7429$

Note: $\text{gcd}$ and $\text{lcm}$ denote the greatest common divisor and lowest common multiple respectively.