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Difference between revisions of "1995 AHSME Problems/Problem 24"

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==Problems==
 
==Problems==
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There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that
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There exist positive integers <math>A,B</math> and <math>C</math>, with no [[greatest common divisor|common factor]] greater than <math>1</math>, such that
  
 
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath>
 
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath>
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==Solution==
 
==Solution==
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<math>A \log_{200} 5 + B \log_{200} 2 = C</math>
+
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath>
  
−
Simplifying and taking the logs away,
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Simplifying and taking the [[logarithm]]s away,
  
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<math>5^A*2^B=200^C=2^{4C}*5^{3C}</math>.
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<cmath>5^A \cdot 2^B=200^C=2^{3C} \cdot 5^{2C}</cmath>
  
−
Therefore, <math>A=3C</math> and <math>B=4C</math>. Since A, B, and C are relatively prime, C=1, B=4, A=3.
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Therefore, <math>A=2C</math> and <math>B=3C</math>. Since <math>A, B,</math> and <math>C</math> are relatively prime, <math>C=1</math>, <math>B=3</math>, <math>A=2</math>. <math>A+B+C=6 \Rightarrow \mathrm{(A)}</math>
  
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<math>A+B+C=8 \Rightarrow \mathrm{(C)}</math>
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==See also==
 +
{{Old AMC12 box|year=1995|num-b=23|num-a=25}}
  
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==See Also==
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[[Category:Introductory Algebra Problems]]
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{{Old AMC12 box|year=1995|num-b=23|num-a=25}}
 

Revision as of 19:08, 7 January 2008

Problems

There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that

\[A \log_{200} 5 + B \log_{200} 2 = C\]

What is $A + B + C$?


$\mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 }$

Solution

\[A \log_{200} 5 + B \log_{200} 2 = C\]

Simplifying and taking the logarithms away,

\[5^A \cdot 2^B=200^C=2^{3C} \cdot 5^{2C}\]

Therefore, $A=2C$ and $B=3C$. Since $A, B,$ and $C$ are relatively prime, $C=1$, $B=3$, $A=2$. $A+B+C=6 \Rightarrow \mathrm{(A)}$

See also

1995 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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
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