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, 2014

Number theory Difficulty 4.3 AIME Find the answer United States

Problem:
What is the smallest positive integer $n$ which cannot be written in any of the following forms?
- $n=1+2+\cdots+k$ for a positive integer $k$ .
- $n=p^{k}$ for a prime number $p$ and integer $k$ .
- $n=p+1$ for a prime number $p$ .
- $n=p q$ for some distinct prime numbers $p$ and $q$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Answer: 40 The first numbers which are neither of the form $p^{k}$ nor $p q$ are $12,18,20,24,28,30,36,40, \ldots$ . Of these $12,18,20,24,30$ are of the form $p+1$ and 28,36 are triangular. Hence the answer is 40 .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.