Difference between revisions of "1994 AHSME Problems/Problem 28"
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The first case gives <math>c-3=12 \implies c=15</math> and the second case gives <math>c-3=4 \implies c=7</math>, and both of these satisfy all the conditions of the problem, so the number of solutions is <math>\boxed{\textbf{(C) } 2}</math>. | The first case gives <math>c-3=12 \implies c=15</math> and the second case gives <math>c-3=4 \implies c=7</math>, and both of these satisfy all the conditions of the problem, so the number of solutions is <math>\boxed{\textbf{(C) } 2}</math>. | ||
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+ | ==See Also== | ||
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+ | {{AHSME box|year=1994|num-b=27|num-a=29}} | ||
+ | {{MAA Notice}} |
Revision as of 16:29, 9 January 2021
Problem
In the -plane, how many lines whose -intercept is a positive prime number and whose -intercept is a positive integer pass through the point ?
Solution
Let the line be , with being a positive integer. Then we have so . Now the -intercept is
We need the -intercept to be positive, so and together imply , so if the -intercept is to be an integer, must be a positive factor of . Hence we can get , , , , , and , and the only ones of these that are prime are and .
The first case gives and the second case gives , and both of these satisfy all the conditions of the problem, so the number of solutions is .
See Also
1994 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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