2023 AMC 10A Problem 12

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

The reversed number ends in the first digit of

N,

and for divisibility by

5

that digit must be

0

or

5

Since

N

is three digits it starts with

5;

count the multiples of

7

from

500

to

599

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.

Problem 12 in other years