2005 AMC 12B Problems/Problem 17: Difference between revisions

Revision as of 17:18, 21 February 2010

How many distinct four-tuples $(a,b,c,d)$ of rational numbers are there with

$a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?$

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 17 \qquad \mathrm{(D)}\ 2004 \qquad \mathrm{(E)}\ \text{infinitely many}$

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 == Problem ==
+How many distinct four-tuples <math>(a,b,c,d)</math> of rational numbers are there with
+<cmath>a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?</cmath>
+<math>
+\mathrm{(A)}\ 0      \qquad
+\mathrm{(B)}\ 1      \qquad
+\mathrm{(C)}\ 17     \qquad
+\mathrm{(D)}\ 2004   \qquad
+\mathrm{(E)}\ \text{infinitely many}
+</math>
 == Solution ==