# Remove the digit

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Let $n$ be a five digit number ,whose first digit is non-zero, and let $m$ be the four digit number formed from n by removing its middle digit. Determine all $n$ such that $n/m$ is an integer

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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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