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2007 AMC 10A Problem 9

Problem 9 of 25EasierAlgebraArithmetic

Real numbers aa and bb satisfy the equations 3a=81b+23^a = 81^{b+2} and 125b=5a−3.125^b = 5^{a-3}. What is ab?ab?

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Solution

The equations become 3a=34(b+2)3^a = 3^{4(b+2)} and 53b=5a−3.5^{3b} = 5^{a-3}. So a=4(b+2)a = 4(b + 2) and 3b=a−3.3b = a - 3. Solving gives a=−12a = -12 and b=−5,b = -5, so ab=60.ab = 60. Thus, the correct answer is E.
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