Difference between revisions of "1970 Canadian MO Problems/Problem 4"
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Revision as of 00:15, 3 May 2020
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
Since x has to be an integer, n has to be greater than or equal to 3. Thus, the answer is
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
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.