, 2003

Number theory Difficulty 5.0 AIME, harder Prove it Italy

Problem:

We take a positive integer $n$ , we sum its digits, and then we again sum the digits of that sum, obtaining an integer $S$ . What is the smallest $n$ that allows obtaining $S \geq 10$ ?

Solution

Solution:

The answer is 199. Indeed, if $n$ had 1 or 2 digits, $n \leq 99$ , and then the sum of its digits would be $\leq 18$ and the sum of the digits of the number equal to the sum of its digits would be $\leq 9$ , so $n$ has at least three digits. We observe that for $n \leq 198$ , the sum of the digits of the number is still $\leq 18$ . Instead if $n=199$ , the sum of its digits is 19 and the sum of the digits of 19 is 10, as required.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.