2001 AIME I Problems/Problem 8
Call a positive integer a 7-10 double if the digits of the base- representation of form a base- number that is twice . For example, is a 7-10 double because its base- representation is . What is the largest 7-10 double?
We let ; we are given that
(This is because the digits in ' s base 7 representation make a number with the same digits in base 10 when multiplied by 2)
Expanding, we find that
Since the s are base- digits, it follows that , and the LHS is less than or equal to . Hence our number can have at most digits in base-. Letting , we find that is our largest 7-10 double.
Solution 2 (Guess and Check)
Let be the base representation of our number, and let be its base representation.
Given this is an AIME problem, . If we look at in base , it must be equal to , so when is looked at in base
If in base is less than , then as a number in base must be less than .
is non-existent in base , so we're gonna have to bump that down to .
This suggests that is less than .
Guess and check shows that , and checking values in that range produces .
Since this is an AIME problem, the maximum number of digits the 7-10 double can have is 3. Let the number be in base 7. Then the number in expanded form is in base 7 and in base 10. Since the number in base 7 is half the number in base 10, we get the following equation. , which simplifies to . The largest possible value of a is 6 because the number is in base 7. Then to maximize the number, is and is . Therefore, the largest 7-10 double is 630 in base 7, or in base 10.
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