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GRE Prep Club Team Member
Joined: 20 Feb 2017
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The function f is defined for each positive three-digit integer n by
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05 Nov 2021, 07:52
05 Nov 2021, 07:52
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Question Stats:
50% (01:43) correct
50% (02:19) wrong based on 6 sessions
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The function f is defined for each positive three-digit integer n by \(f(n) = 2^x3^y5^z\) , 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 \(f(m)=9*f(v)\) , then \(m-v=\) ?
(A) 8
(B) 9
(C) 18
(D) 20
(E) 80
(A) 8
(B) 9
(C) 18
(D) 20
(E) 80
Re: The function f is defined for each positive three-digit integer n by
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08 Nov 2021, 14:01
08 Nov 2021, 14:01
Expert Reply
Let the 3-digit positive integer \({m}\) be \({abc}\) and positive integer \(v\) be \(rst\). Question: \((100a+10b+c)-(100r+10s+t)=?\).
Then \(f(m)=2^a*3^b*5^c\) and \(f(v)=2^r*3^s*5^t\).
Given: \(f(m)=9*f(v)\) --> \(2^a*3^b*5^c=9*2^r*3^s*5^t\) --> \(2^{a-r}*3^{b-s}*5^{c-t}=3^2\) (or \(2^a*3^b*5^c=2^r*3^{s+2}*5^t\)).
So, \(a-r=0\), \(b-s=2\) and \(c-t=0\) (or \(a=r\), \(b=s+2\) and \(c=t\)) --> \(m-v=(100a+10b+c)-(100r+10s+t)=10b-10(b-2)=20\). For example if \(m\) and \(v\) are 143 and 123 respectively (hundreds and units digit are equal and tens digit of \(m\) is 2 more than tens digit of \(v\)) then \(143-123=20\).
Answer: D.
Then \(f(m)=2^a*3^b*5^c\) and \(f(v)=2^r*3^s*5^t\).
Given: \(f(m)=9*f(v)\) --> \(2^a*3^b*5^c=9*2^r*3^s*5^t\) --> \(2^{a-r}*3^{b-s}*5^{c-t}=3^2\) (or \(2^a*3^b*5^c=2^r*3^{s+2}*5^t\)).
So, \(a-r=0\), \(b-s=2\) and \(c-t=0\) (or \(a=r\), \(b=s+2\) and \(c=t\)) --> \(m-v=(100a+10b+c)-(100r+10s+t)=10b-10(b-2)=20\). For example if \(m\) and \(v\) are 143 and 123 respectively (hundreds and units digit are equal and tens digit of \(m\) is 2 more than tens digit of \(v\)) then \(143-123=20\).
Answer: D.
gmatclubot
Re: The function f is defined for each positive three-digit integer n by [#permalink]
08 Nov 2021, 14:01
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