2023 AMC 10A Problem 12

Problem 12 of 25IntermediateNumber Theory Combinatorics

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.

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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

5.

A three-digit number can’t start with

0,

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

7 \cdot 85 = 595,

that’s

14

numbers. Therefore, the answer is B.

Tagged: divisibility · digits · counting integers in a range