1971 Canadian MO Problems/Problem 7
Let be a five digit number (whose first digit is non-zero) and let be the four digit number formed from n by removing its middle digit. Determine all such that is an integer.
Let and , where , , , , and are base-10 digits and . If is an integer, then , or
This implies that
Clearly we have that , as is positive. Therefore, this quotient must be equal to 9 (note that this does not mean ), and
This simplifies to . The only way that this could happen is that . Then . Therefore the only values of such that is an integer are multiples of 1000. It is not hard to show that these are all acceptable values.
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