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>. | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
