Difference between revisions of "2001 AIME I Problems/Problem 8"
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== Problem == | == Problem == | ||
− | Call a positive integer <math>N</math> a | + | Call a positive integer <math>N</math> a ''7-10 double'' if the digits of the base-<math>7</math> representation of <math>N</math> form a base-<math>10</math> number that is twice <math>N</math>. For example, <math>51</math> is a 7-10 double because its base-<math>7</math> representation is <math>102</math>. What is the largest 7-10 double? |
== Solution == | == Solution == | ||
− | {{ | + | We let <math>N_7 = \overline{a_na_{n-1}\cdots a_0}_7</math>; we are given that |
+ | |||
+ | <cmath>2(a_na_{n-1}\cdots a_0)_7 = (a_na_{n-1}\cdots a_0)_{10}</cmath> | ||
+ | |||
+ | Expanding, we find that | ||
+ | |||
+ | <cmath>2 \cdot 7^n a_n + 2 \cdot 7^{n-1} a_{n-1} + \cdots + 2a_0 = 10^na_n + 10^{n-1}a_{n-1} + \cdots + a_0</cmath> | ||
+ | |||
+ | or re-arranging, | ||
+ | |||
+ | <cmath>a_0 + 4a_1 = 2a_2 + 314a_3 + \cdots + (10^n - 2 \cdot 7^n)a_n</cmath> | ||
+ | |||
+ | Since the <math>a_i</math>s are base-<math>7</math> digits, it follows that <math>a_i < 7</math>, and the LHS is less than <math>30</math>. Hence our number can have at most <math>3</math> digits in base-<math>7</math>. Letting <math>a_2 = 6</math>, we find that <math>630_7 = \boxed{315}_{10}</math> is our largest 7-10 double. | ||
== See also == | == See also == | ||
{{AIME box|year=2001|n=I|num-b=7|num-a=9}} | {{AIME box|year=2001|n=I|num-b=7|num-a=9}} | ||
+ | |||
+ | [[Category:Intermediate Number Theory Problems]] |
Revision as of 22:28, 11 June 2008
Problem
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?
Solution
We let ; we are given that
Expanding, we find that
or re-arranging,
Since the s are base-
digits, it follows that
, and the LHS is less than
. Hence our number can have at most
digits in base-
. Letting
, we find that
is our largest 7-10 double.
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |