Mock AIME 2 2006-2007 Problems/Problem 1
Contents
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 .
Solution2
Note: a+b+c+d=a+a+8+(a+4)/4+4a+16 =(25a+100)/4 = 25(a+4)/4 . Therefore, we know that a+4/4 has to be an integer. since a>0, the smaller integer we can have is 2, so a=4. Therefore, the smallest value is 50.
~Please convert to latex
See Also
Mock AIME 2 2006-2007 (Problems, Source) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |