2020 AIME I Problems/Problem 3
Contents
Problem
A positive integer has base-eleven representation
and base-eight representation
where
and
represent (not necessarily distinct) digits. Find the least such
expressed in base ten.
video solution
Solution
From the given information, . Since
,
, and
have to be positive,
. Since we need to minimize the value of
, we want to minimize
, so we have
. Then we know
, and we can see the only solution is
,
. Finally,
, so our answer is
.
~ JHawk0224
Solution 2 (Official MAA)
The conditions of the problem imply that , so
. The maximum digit in base eight is
and because
, it must be that
is
or
When
, it follows that
, which implies that
. Then
must be
or
If
, then
is not an integer, and if
, then
, so
. Thus
, and
. The number
also satisfies the conditions of the problem, but
is the least such number.
Video Solution:
Minor edits by TryhardMathlete
Video Solution
https://youtu.be/mgEZOXgIZXs?t=1204
~ pi_is_3.14
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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