Difference between revisions of "2006 Alabama ARML TST Problems/Problem 5"
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==Problem== | ==Problem== | ||
− | There exist positive integers <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> with no common factor greater than 1, such that | + | There exist positive integers <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> with no [[common factor]] greater than 1, such that |
<center><math>A\log_{1200} 2+B\log_{1200} 3+C\log_{1200} 5=D.</math></center> | <center><math>A\log_{1200} 2+B\log_{1200} 3+C\log_{1200} 5=D.</math></center> |
Revision as of 11:59, 17 April 2008
Problem
There exist positive integers ,
,
, and
with no common factor greater than 1, such that
![$A\log_{1200} 2+B\log_{1200} 3+C\log_{1200} 5=D.$](http://latex.artofproblemsolving.com/a/6/d/a6da2ed10da3db30008df596117c45ab02327603.png)
Find .
Solution
Simplifying and taking the logarithms away,
Therefore, ,
, and
. Since
and
are relatively prime,
,
,
,
.
See also
2006 Alabama ARML TST (Problems) | ||
Preceded by: Problem 4 |
Followed by: Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |