Difference between revisions of "1962 IMO Problems/Problem 1"
(New page: ==Problem== Find the smallest natural number <math>n</math> which has the following properties: (a) Its decimal representation has 6 as the last digit. (b) If the last digit 6 is erased ...) |
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==Solution== | ==Solution== | ||
| − | { | + | As the new number starts with a <math>6</math> and the old number is <math>1/4</math> of the new number, the old number must start with a <math>1</math>. |
| + | |||
| + | As the new number now starts with <math>61</math>, the old number must start with <math>\lfloor 61/4\rfloor = 15</math>. | ||
| + | |||
| + | We continue in this way until the process terminates with the new number <math>615\,384</math> and the old number <math>n=\boxed{153\,846}</math>. | ||
==See Also== | ==See Also== | ||
{{IMO box|year=1962|before=First Question|num-a=2}} | {{IMO box|year=1962|before=First Question|num-a=2}} | ||
Revision as of 16:22, 31 January 2009
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 .
Solution
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 
.
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 | ||
