AMC 10 A

Algebra Difficulty 3.8 AMC 10/12 Find the answer United States

The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?

Your answer

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

Solution

In order to minimize the number of terms in the sum, the greatest two-digit number, $99$ , should be used as many times as possible. Because $20 \cdot 99 = 1980$ , the minimum number of terms is greater than $20$ . On the other hand, $2024 = 20 \cdot 99 + 44$ , so the least number of two-digit numbers needed is $21$ .

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.