PUMaC 2023 · 数论（B 组） · 第 2 题

PUMaC 2023 — Number Theory (Division B) — Problem 2

I have a four-digit palindrome a b b a that is divisible by b and is also divisible by the two-digit number b b . Find the number of palindromes satisfying both of these properties. 3 3 3

解析

I have a four-digit palindrome a b b a that is divisible by b and is also divisible by the two-digit number b b . Find the number of palindromes satisfying both of these properties. Proposed by Austen Mazenko Answer: 31 Notice the second condition implies the first condition. Now we do casework: note b b | a b b a = ⇒ b b | a 0 0 a = ⇒ 11 b | a · 7 · 11 · 13. Thus, we just need b | a · 7 because b < 10 and 13 is prime. Doing casework on all the choices b < 10, we get a total of 9 + 4 + 3 + 2 + 1 + 1 + 9 + 1 + 1 = 31. 3 3 3