not really elegant (it is more intuitive).anyways, my approach-
5^x-5^y=2^y/2*5^x/5
10(5^x-5^y)=2^y*5^x
I picked answer choice E 12,since it appears to be the easiest one.
12=3*4
let x=4 y=3
10(5^4-5^3)=2^3*5^4
5^3*2*5(5-1)=2^3*5^4
5^3*2^3*5=2^3*5^3*5
Algebraic approach to this one
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Since 5^(x-1) is a factor of the righthand side, 5^(x-1) must be a factor of the lefthand side.If x and y are positive integers and 5^x - 5^y = 2^(y-1) * 5^(x-1), what is xy?
A)48 B)36 C)24 D)18 E)12
(5^x - 5^y) / 5^(x-1) = 2^(y-1) * 5^(x-1) / 5^(x-1) Divide each side by 5^(x-1).
5^(x-x+1) - 5^(y-x+1) = 2^(y-1) Subtract the exponents on the lefthand side and simplify the righthand side.
5 - 5^(y-x+1) = 2^(y-1).
Since the righthand side is a positive integer, so must be the lefthand side.
Thus, we know that 5^(y-x+1) = 5� = 1.
Otherwise, the lefthand side will not be a positive integer.
Substituting 5^(y-x+1)=1 into 5 - 5^(y-x+1) = 2^(y-1), we get:
5-1 = 2^(y-1)
4 = 2^(y-1)
y=3.
Since y-x+1=0, we get:
3-x+1 = 0
x=4.
Thus, xy = 4*3 = 12.
The correct answer is E.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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