Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | A [[positive integer]] is called a ''dragon'' if it can be written as the sum of four positive integers <math> | + | A [[positive integer]] is called a ''dragon'' if it can be written as the sum of four positive integers <math>a,b,c,</math> and <math>d</math> such that <math>a+4=b-4=4c=d/4.</math> Find the smallest dragon. |
==Solution== | ==Solution== | ||
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| − | *[[Mock AIME 2 2006-2007/Problem 2 | Next Problem]] | + | *[[Mock AIME 2 2006-2007 Problems/Problem 2 | Next Problem]] |
*[[Mock AIME 2 2006-2007]] | *[[Mock AIME 2 2006-2007]] | ||
Revision as of 15:35, 3 April 2012
Problem
A positive integer is called a dragon if it can be written as the sum of four positive integers 
and 
such that 
Find the smallest dragon.
Solution
From 
we have that 16 divides . From 
we have 
. Minimizing minimizes 
and 
and consequently minimizes our dragon. The smallest possible choice is 
, from which 
and 
so our desired number is 
.
