Difference between revisions of "AoPS Wiki talk:Problem of the Day/July 21, 2011"
(→Solution) |
m (→Solution) |
||
| Line 2: | Line 2: | ||
{{:AoPSWiki:Problem of the Day/July 21, 2011}} | {{:AoPSWiki:Problem of the Day/July 21, 2011}} | ||
==Solution== | ==Solution== | ||
| − | Note: someone check | + | Note: someone check the arithmetic please |
For purposes of generalization, let the equations be <math>ax+by=1002001</math> and <math>cx+dy=2004002</math>. Notice that <math>2*1002001 = 2004002</math>. Swap this out for a new variable <math>z</math>. | For purposes of generalization, let the equations be <math>ax+by=1002001</math> and <math>cx+dy=2004002</math>. Notice that <math>2*1002001 = 2004002</math>. Swap this out for a new variable <math>z</math>. | ||
Revision as of 15:57, 21 July 2011
Problem
AoPSWiki:Problem of the Day/July 21, 2011
Solution
Note: someone check the arithmetic please
For purposes of generalization, let the equations be 
and 
. Notice that 
. Swap this out for a new variable 
.
This gives 
.
Multiply the left equation by two and substitute it into the other equation.
, which implies that
. Substituting the actual numbers back in gives:
Simplifying:
Which further simplifies to:
Therefore:
Sample solution: 
General Solution: 
