Difference between revisions of "1985 AHSME Problems/Problem 26"
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Find the least [[positive integer]] <math> n </math> for which <math> \frac{n-13}{5n+6} </math> is a non-zero reducible fraction. | Find the least [[positive integer]] <math> n </math> for which <math> \frac{n-13}{5n+6} </math> is a non-zero reducible fraction. | ||
| − | <math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\ | + | <math> \mathrm{(A)\ } 45 \qquad \mathrm{(B) \ }68 \qquad \mathrm{(C) \ } 155 \qquad \mathrm{(D) \ } 226 \qquad \mathrm{(E) \ }\text84 </math> |
==Solution== | ==Solution== | ||
Revision as of 10:01, 23 June 2017
Problem
Find the least positive integer 
for which 
is a non-zero reducible fraction.
Solution
For the fraction to be reducible, the greatest common factor of the numerator and the denominator must be greater than 
. By the Euclidean algorithm,
Since 
is prime, 
must be a multiple of , which first occurs when 
.
See Also
| 1985 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 25 |
Followed by Problem 27 | |
| 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. 
