Question

How many three-digit positive integers $N$ satisfy the following properties?

• The number $N$ is divisible by $7.$

• The number formed by reversing the digits of $N$ is divisible by $5.$

$13$

$14$

$15$

$16$

$17$

Answer: B

Concepts:divisibility digits counting integers in a range

Difficulty rating: 1440

Solution:

When we reverse $N,$ its last digit is the first digit of $N.$ For the reversal to be divisible by $5,$ that digit is $0$ or $5.$ A three-digit number can't start with $0,$ so $N$ starts with $5,$ meaning $500 \le N \le 599$ (and the reversal ends in $5,$ always fine). Now just count multiples of $7$ here: from $7 \cdot 72 = 504$ to $7 \cdot 85 = 595,$ that's $14$ numbers. Therefore, the answer is B.

2023 AMC 10A Problem 12

Answer: B

Difficulty rating: 1440

Solution:

Problem 12 in Other Years