Problem:
The decimal representation of , where and are relatively prime positive integers and , contains the digits , and consecutively, and in that order. Find the smallest value of for which this is possible.
Solution:
To find the smallest , it is sufficient to consider the case in which the string occurs immediately after the decimal point. To show this, suppose that in the decimal representation of , the string does not occur immediately after the decimal point. Then , where represents a block of digits, . This implies that , but , which is between and , can then be expressed in the form , where and are relatively prime positive integers and . Now
It follows that . The remainder when is divided by must therefore be less than , so it is sensible to investigate multiples of that are close to and less than a multiple of . When yields as the multiple of that is closest to and less than ; but the remainder is greater than . When yields as the multiple of that is closest to and less than ; but the remainder is greater than . More generally, is less than when , and the remainder is . The remainder is less than when , that is, when . Thus the minimum value of is , and the minimum value of is .
The problems on this page are the property of the MAA's American Mathematics Competitions