Question

The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers?

$40\text{ and }50$

$51\text{ and }55$

$56\text{ and }60$

$61\text{ and }65$

$66\text{ and }99$

Answer: D

Concepts:least common multiple modular arithmetic

Difficulty rating: 1240

Video solution:

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Written solution:

Let the number be $x.$ Since it leaves a remainder of $2$ when divided by $3,4,5,6,$ we know $x-2$ is a multiple of $3,4,5,6.$ This means $x-2$ is a multiple of $lcm(3,4,5,6)$ which is $60.$ Therefore, $x-2$ must be a multiple of $60.$ The next number such that this occurs is when $x-2 = 60 \implies x = 62 .$

Thus, the answer is D .

2012 AMC 8 Problem 15

Answer: D

Difficulty rating: 1240

Video solution:

Written solution:

Problem 15 in Other Years