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== |
Latest revision as of 14:28, 22 June 2018
Problem
Find the number of lattice points that the line passes through in Quadrant I.
Solution
Solve for to get 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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