Question

How many positive three-digit integers have a remainder of $2$ when divided by $6,$ a remainder of $5$ when divided by 9, and a remainder of $7$ when divided by $11?$

$1$

$2$

$3$

$4$

$5$

Answer: E

Concepts:Chinese Remainder Theorem least common multiple counting integers in a range

Difficulty rating: 1490

Video solution:

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Written solution:

Each required remainder is $4$ less than its divisor, so $x+4$ must be divisible by $6,9,$ and $11$ . Hence $x+4$ is a multiple of $\operatorname{lcm}(6,9,11)=198.$

For a three-digit $x$ , we have $104\le x+4\le1003$ . The multiples of $198$ in this interval are $198,396,594,792,990$ , giving $5$ possible integers.

Thus, E is the correct answer.

2018 AMC 8 Problem 21

Answer: E

Difficulty rating: 1490

Video solution:

Written solution:

Problem 21 in Other Years