1962 IMO Problems/Problem 1
Contents
[hide]Problem
Find the smallest natural number 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 .
Video Solution
https://youtu.be/9y5UUNIhUfU?si=PzXbNokxOXCRxYBh [Video Solution by little-fermat]
Solution 1
As the new number starts with a and the old number is
of the new number, the old number must start with a
.
As the new number now starts with , the old number must start with
.
We continue in this way until the process terminates with the new number and the old number
.
Solution 2
We know from the two properties that for some string ,
. Let the number of digits in
be
; then moving the
to the front would give it place value
; as a result,
. Multiplying this by
gives
, and subtracting the former yields
, or
. As a result,
. By Fermat's Little Theorem, we know that
divides
, so it isn't difficult to try values of
less than
to find the smallest such
.
Eventually, we notice that divides
, so
. Then
, and since the number ends in
, we know that
is also an integer, so this is the solution.
~eevee9406
See Also
1962 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |