Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 1"
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= 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. | = 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. | Therefore, the smallest value is 50. | ||
| + | |||
| + | ~Please convert to latex | ||
==See Also== | ==See Also== | ||
Latest revision as of 00:27, 11 April 2024
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 | ||
