Difference between revisions of "1995 AHSME Problems/Problem 24"
(New page: ==Problems== There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that <cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath> What is...) |
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Simplifying and taking the logs away, | Simplifying and taking the logs away, | ||
− | <math>5^A*2^B=200^C=2^4C*5^3C</math>. | + | <math>5^A*2^B=200^C=2^{4C}*5^{3C}</math>. |
Therefore, <math>A=3C</math> and <math>B=4C</math>. Since A, B, and C are relatively prime, C=1, B=4, A=3. | Therefore, <math>A=3C</math> and <math>B=4C</math>. Since A, B, and C are relatively prime, C=1, B=4, A=3. |
Revision as of 11:32, 7 January 2008
Problems
There exist positive integers and , with no common factor greater than 1, such that
What is ?
Solution
Simplifying and taking the logs away,
.
Therefore, and . Since A, B, and C are relatively prime, C=1, B=4, A=3.