PUMaC 2012 · 数论（B 组） · 第 1 题

PUMaC 2012 — Number Theory (Division B) — Problem 1

[ 3 ] When some number a is written in base b , the result is 144 . a and b also happen to be b integer side lengths of a right triangle. If a and b are both less than 20, find the sum of all possible values of a .

解析

[ 3 ] When some number a is written in base b , the result is 144 . a and b also happen to be b integer side lengths of a right triangle. If a and b are both less than 20, find the sum of all possible values of a . 2 2 Solution: 144 = b + 4 b + 4 = ( b + 2) . Therefore, a = b + 2. (Thanks to multiple teams b for pointing out the following fact:) If for example we make a and b the lengths of the legs, √ 2 2 then the hypoteneuse can always exist with length c = a + b , which does not necessarily need to be an integer. Thus all possible values of a, b such that b > 4 (for 144 to be a possible b number) and a, b < 20 are possible, so the possible values of a are all integers from 7 to 19, (7+19)(13) and the answer is = 169 . 2 Problem contributed by Elizabeth Yang