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# Indices and Algebra

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Manager
Joined: 02 Sep 2008
Posts: 102

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17 Mar 2009, 21:12
The function f is defined for each positive three-digit integer n by f(n) = 2^x*3^y*5^z, where x, y and z
are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive
integers such that f(m) = 9f(v), then m-v = ?
A. 8
B. 9
C. 18
D. 20
E. 80

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Director
Joined: 14 Aug 2007
Posts: 695

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17 Mar 2009, 22:08
milind1979 wrote:
The function f is defined for each positive three-digit integer n by f(n) = 2^x*3^y*5^z, where x, y and z
are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive
integers such that f(m) = 9f(v), then m-v = ?
A. 8
B. 9
C. 18
D. 20
E. 80

D.20
lets m = abc and v= efg
f(m)= 2^a*3^b*5^c
f(v) = 2^e*3^f*5^g

f(m)= 9*f(v) = 3^2*(2^e*3^f*5^g) = 2^e*3^f+2*5^g

2^a*3^b*5^c = 2^e*3^f+2*5^g

=> units digits of m & v are same, and 100s digit of m & v are same, b = f+2 i.e only the 10s digit is of m is greater than v by 2
so we have a(f+2)c - a f c = 20
Manager
Joined: 02 Mar 2009
Posts: 122

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17 Mar 2009, 22:54
Got D as well.
Took the same route.
Senior Manager
Joined: 30 Nov 2008
Posts: 479
Schools: Fuqua

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19 Mar 2009, 10:10
Agree with D. But to simplyfy the solution, by reducing the number of variables,

Let V be abc. Then f(v) = $$2^a * 3^b * 5^c$$.
Given, f(m) = 9f(v) ==> f(m) = $$2^a$$ * $$3^(b+2)$$ * $$5^c$$==> m = a(b+2)c.

Note: - Here m and v are expressed in terms face value.

m-v = a(b+2)c - abc. Consider the place value of each digit.

(100a + 10(b+2) + c) - (100a + 10b + c). Solving this we get 20.

So m-v = 20.

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Re: Indices and Algebra   [#permalink] 19 Mar 2009, 10:10
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# Indices and Algebra

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