PUMaC 2017 · 数论（A 组） · 第 3 题
PUMaC 2017 — Number Theory (Division A) — Problem 3
题目详情
- Define the bigness of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer N for which there exists a rectangular prism with integer side lengths and bigness N and another one with integer side lengths and bigness N + 1.
解析
- The content of a rectangular prisms with side lengths a , b , and c is ( a + 2)( b + 2)( c + 2) − 8 , so there is a rectangular prism with integer side lengths and content N iff N + 8 is a product of three integers, each of which is greater than 2. A bit of case work shows the least such N is 55 . Problem written by Matt Tyler ′ ′