Track / Stage 4 / 295 of 340 #555 of 1964

Problem 555

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

Let's take a positive integer $n$ , sum its digits, and then add the digits of this sum again to get an integer $S$ . What is the smallest $n$ that allows us to obtain $S \geq 10?$

Your answer

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

Official solution

11. 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. Note that for $n \leq 198$ , the sum of the digits of the number is still $\leq 18$ . However, if $n=199$ , the sum of its digits is 19 and the sum of the digits of 19 is 10, as required.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.