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jmc

number theory intermediate

Problem

The base-10 integers 36, 64, and 81 can be converted into other bases so that their values are represented by the same digits , where and are two distinct digits from 0-9. What is the value of ?
Solution
36, 64, and 81 are all perfect squares. That means they can all be written as , where is the square root of each number. So, all three numbers can be represented by the digits when converted to other bases. To see that no other digits work, note that only the bases 4, 5, and 6 use three digits to represent the number 36. (This follows from , which expresses the condition that 36 has 3 digits in base ). The representations are , and , only one of which fits the form .
Final answer
100