# 1962 IMO Problems/Problem 1

## Problem

Find the smallest natural number $n$ which has the following properties:

(a) Its decimal representation has 6 as the last digit.

(b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

## Video Solution

https://youtu.be/9y5UUNIhUfU?si=PzXbNokxOXCRxYBh [Video Solution by little-fermat]

## Solution 1

As the new number starts with a $6$ and the old number is $1/4$ of the new number, the old number must start with a $1$.

As the new number now starts with $61$, the old number must start with $\lfloor 61/4\rfloor = 15$.

We continue in this way until the process terminates with the new number $615\,384$ and the old number $n=\boxed{153\,846}$.

## Solution 2

Let the original number = $10n + 6$, where $n$ is a 5 digit number.

Then we have $4(10n + 6) = 600000 + n$.

=> $40n + 24 = 600000 + n$.

=> $39n = 599976$.

=> $n = 15384$.

=> The original number = $\boxed{153\,846}$.