HMMT November

Number theory Difficulty 4.1 AIME Find the answer United States

Problem:

How many perfect squares divide $10^{10}$ ?

Your answer

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

Solution

Solution:

Answer: 36
A perfect square $s$ divides $10^{10}$ if and only if $s = 2^{a} \cdot 5^{b}$ where $a, b \in \{0, 2, 4, 6, 8, 10\}$ . There are 36 choices, giving 36 different $s$ 's.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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