2020 AIME I Problems/Problem 10: Difference between revisions
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== Problem == | == Problem == | ||
Let <math>m</math> and <math>n</math> be positive integers satisfying the conditions | |||
<math>\quad\bullet\ \gcd(m+n,210)=1,</math> | |||
<math>\quad\bullet\ m^m</math> is a multiple of <math>n^n,</math> and | |||
<math>\quad\bullet\ m</math> is not a multiple of <math>n.</math> | |||
Find the least possible value of <math>m+n.</math> | |||
== Solution == | == Solution == | ||
Revision as of 16:35, 12 March 2020
Note: Please do not post problems here until after the AIME.
Problem
Let 
and 
be positive integers satisfying the conditions
is a multiple of 
and
is not a multiple of 
Find the least possible value of 
Solution
Taking inspiration from 
we are inspired to take to be 
, the lowest prime not dividng 
, or 
. Now, there are 
factors of 
, so 
, and then 
for 
. Now, 
. Noting 
is the minimal that satisfies this, we get 
. Thus, it is easy to verify this is minimal and we get 
. ~awang11
See Also
| 2020 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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