Difference between revisions of "1985 AHSME Problems/Problem 26"
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<math> \gcd(71, n-13) </math> | <math> \gcd(71, n-13) </math> | ||
| − | Since <math> 71 </math> is prime, <math> n-13 </math> must be a multiple of <math> 71 </math>, which first occurs when <math> n=71+13=84, \boxed{\text{E}} </math>. | + | Since <math> 71 </math> is prime, <math> n-13 </math> must be a multiple of <math> 71 </math>, which first occurs when <math> n=71+13=84</math>, <math>\boxed{\text{(E) 84}} </math>. |
==See Also== | ==See Also== | ||
{{AHSME box|year=1985|num-b=25|num-a=27}} | {{AHSME box|year=1985|num-b=25|num-a=27}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 17:07, 9 May 2020
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. 
