Difference between revisions of "2008 iTest Problems/Problem 18"
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In order for <math>y</math> to be an positive integer, <math>1909-19x</math> must be a multiple of 20 greater than <math>0</math>, so <math>x \le 100</math>. This means that the ones digit of <math>1909-19x</math> is <math>0</math> and the tens digit of <math>1909-19x</math> is even. | In order for <math>y</math> to be an positive integer, <math>1909-19x</math> must be a multiple of 20 greater than <math>0</math>, so <math>x \le 100</math>. This means that the ones digit of <math>1909-19x</math> is <math>0</math> and the tens digit of <math>1909-19x</math> is even. | ||
| − | The ones digit of <math>1909-19x</math> is <math>0</math> when the last digit of <math>x</math> is <math>1</math>, so the available options are <math>1, 11, 21 \cdots 91</math>. However, since <math>1909-19x=1909-20x+x</math>, the tens digit must be odd. Thus, the only values that work are <math>11</math>, <math>31</math>, <math>51</math>, <math>71</math>, and <math>91</math>, so there are only <math>5</math> lattice points in the first quadrant. | + | The ones digit of <math>1909-19x</math> is <math>0</math> when the last digit of <math>x</math> is <math>1</math>, so the available options are <math>1, 11, 21 \cdots 91</math>. However, since <math>1909-19x=1909-20x+x</math>, the tens digit must be odd. Thus, the only values that work are <math>11</math>, <math>31</math>, <math>51</math>, <math>71</math>, and <math>91</math>, so there are only <math>\boxed{5}</math> lattice points in the first quadrant. |
==See Also== | ==See Also== | ||
Revision as of 13:28, 22 June 2018
Problem
Find the number of lattice points that the line 
passes through in Quadrant I.
Solution
Solve for 
to get
![\[y = \frac{1909-19x}{20}\]](http://latex.artofproblemsolving.com/4/f/a/4fa2409fa20b7df2542d2b9c74487341eb5f3fa5.png)
In order for to be an positive integer, 
must be a multiple of 20 greater than 
, so 
. This means that the ones digit of is and the tens digit of is even.
The ones digit of is when the last digit of 
is 
, so the available options are 
. However, since 
, the tens digit must be odd. Thus, the only values that work are 
, 
, 
, 
, and 
, so there are only 
lattice points in the first quadrant.
See Also
| 2008 iTest (Problems) | ||
| Preceded by: Problem 17 |
Followed by: Problem 19 | |
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