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?

$40$ and $50$

$51$ and $55$

$56$ and $60$

$61$ and $65$

$66$ and $99$

Answer: D

Concepts:least common multiple modular arithmetic

Difficulty rating: 1240

Small Hint:

Subtract $2$ from the unknown number.

Big Hint:

The result must be a common multiple of $3$ , $4$ , $5$ , and $6$ .

Video solution:

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Written solution:

Let the number be $x.$ Since it leaves a remainder of $2$ when divided by $3$ , $4$ , $5$ , and $6,$ we know $x-2$ is a multiple of $3$ , $4$ , $5$ , and $6.$ This means $x-2$ is a multiple of $\operatorname{lcm}(3,4,5,6)$ , 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