PUMaC 2011 · 数论（B 组） · 第 2 题

PUMaC 2011 — Number Theory (Division B) — Problem 2

[ 3 ] Two robots are programmed to communicate numbers using different bases. The first robot states: “I communicate in base 10, which interestingly is a perfect square. You communicate in base 16, which is not a perfect square.” The second robot states: “I find it more interesting that the sum of our bases is the factorial of an integer.” The second robot is referring to the factorial of which integer?

解析

Suppose that Robot 1’s base is b and Robot 2’s base is b . From the first statement, we know 1 2 that the first robot’s base is 10 = b and that the second robot’s base is b = 16 = b + 6. b 1 2 b 1 1 1 From the second statement, we know that b + b = 2 b + 6 = n ! for some integer n . Since b 2 1 1 1 is a perfect square, b ≡ 1 or 4 (mod 5), so n ! ≡ 3 or 4 (mod 5). However, this implies that 1 n ≤ 4. Checking all n ≤ 5 shows that b is a perfect square only when n = 4 ( b = 9), so 1 1 n = 4 . i j 1 2 i 1 2 j