1970 Canadian MO Problems/Problem 4
Problem
Solution
a)
Let the integer have n digits. Then the integer can be represented in the form of 
where x is a non-negative integer less than 
. The problem is asking for integers such as 
. Solving for x results in ![\[x = \frac{10^{n-1}}{4}\]](http://latex.artofproblemsolving.com/7/a/8/7a86ded68e96e8bba2b9c5acb94033f4f9823fa9.png)
Since x has to be an integer, n has to be greater than or equal to 3. Thus, the answer is ![\[625, 6250, 62500, \ldots\]](http://latex.artofproblemsolving.com/4/3/f/43fac99965dfdfa0b924c16c902b0d65033ba6a2.png)
b)
We use the same notation as part a. Then, the condition can be represented as 
where 
is an integer between 1 and 9 inclusive. Solving for x results in ![\[x = \frac{m*10^{n-1}}{34}\]](http://latex.artofproblemsolving.com/f/9/4/f94c788d14e8eedabf8978efe845ce26979d6573.png)
For x to be an integer, the numerator of the expression has to be divisible by 34, and thus 17. obviously does not have any factors of 17, meaning that must have a factor of 17, which is also impossible since m is an integer between 1 and 9. Therefore, an integer required by the problem is not possible.
