Track / Stage 5 / 46 of 400 #646 of 1964

Problem 646

AIME late

Number theory Difficulty 5.2 Find the answer

33.1. A six-digit number starts with one. If this one is moved to the end of the number, the resulting number will be three times the given number. Find the given number.

$\text { (7-10 grades) }$

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

33.1. The desired six-digit number has the form $1 \overline{a b c d e}$ . According to the problem, $3 \cdot 1 \overline{a b c d e}=\overline{a b c d e} 1$ or $3(100000+\overline{a b c d e})=1+10 \cdot \overline{a b c d e}$ . From this, $7 \cdot \overline{a b c d e}=299999$ , i.e., $\overline{a b c d e}=42857$ . Therefore, the desired number is 142857.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.