1970 Canadian MO Problems/Problem 4
a) Find all positive integers with initial digit such that the integer formed by deleting is of the original integer.
b) Show that there is no integer such that the deletion of the first digit produces a result that is of the original integer.
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.
Sol by Vwall