Number Theory

# Greatest Common Divisor / Lowest Common Multiple: Level 1 Challenges

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?

###### Image credit: Wikipedia Pens Pencils

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.

While setting problems on Brilliant, Calvin absentmindedly eats through 7 identical bags of gummy worms. He looks up the 'nutritional information' on the bag, and notices that each bag of gummy worms contains 6 servings, and each serving has a whole number of calories. Given this, Calvin calculated that he consumed just under 1000 calories. What is the maximum possible number of calories that Calvin could have consumed from these bags of gummy worms?

The traffic lights at three different intersections change after every 48 seconds, 72 seconds, and 108 seconds, respectively. If they change simultaneously at 9 a.m., then when is the next time that they change simultaneously?

×