Track / Stage 4 / 25 of 340 #765 of 2444

Problem 765

AMC 12 late, AIME early

Number theory Difficulty 4.1 Find the answer $10^{\text {th }}$ Annual Harvard-MIT Mathematics Tournament · United States

Find the smallest positive integer that is twice a perfect square and three times a perfect cube.

Your answer

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

Official solution

Solution:

Let $n$ be such a number. If $n$ is divisible by $2$ and $3$ exactly $e_{2}$ and $e_{3}$ times, then $e_{2}$ is odd and a multiple of three, and $e_{3}$ is even and one more than a multiple of three. The smallest possible exponents are $n_{2}=3$ and $n_{3}=4$ . The answer is then $2^{3} \cdot 3^{4}=648$ .

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