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Problem 735

AIME late

Number theory Difficulty 5.4 Find the answer

## Task 1 - 140931

We are looking for the smallest natural number $x$ (where $x$ does not necessarily have to be a single digit), which has the following property:

The number $83 \cdot x$ (the product of 83 and $x$ ) has the digit sequence $3 x 8$ (i.e., a 3 is placed in front of the digit or digit sequence of the number $x$ , and an 8 is placed after the so formed digit sequence).

Your answer

Official solution

$x$ is the sought number precisely when 1 is the smallest natural number for which there exists a natural number $n \geq 2$ such that

$83 x=3 \cdot 10^{n}+10 x+8$

or, equivalently, $73 x=3 \cdot 10^{n}+8$ .

If one examines the numbers 308, 3008, 30008, 300008, 3000008 in sequence for divisibility by 73, it turns out that only the number $3000008=73 \cdot 41096$ is divisible.

Therefore, $x=41096$ is the sought number, and it holds that $83 \cdot 41096=3410968$ .

## Solution adopted from $[5]$

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