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The function f is defined for all positive, three-digit integers as

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Bunuel
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The function f is defined for all positive, three-digit integers as [#permalink] New post   25 Apr 2017, 02:14
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The function f is defined for all positive, three-digit integers as \(f(x)=(2^a)(3^b)(5^c)\), where a, b, and c are the hundreds, tens, and units digits, respectively, of x. What is m−n?


(1) \(f(m)=8f(n)\)

(2) \(n=111\)
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Re: The function f is defined for all positive, three-digit integers as [#permalink] New post   25 Apr 2017, 04:13
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S1. f(m)=8f(n)

f(m) = 2^3 f(n)

f(x) = 2^a * 3^b * 5^c

so, we now know that only a is different in f(m) and f(n), b and c are same.
we also know that a in f(m) is 3 greater than a in f(n).

a is the hundreds digit of the number. So, if m=401, n =101 or m=996, n=696

m-n in all case would be 300.

Sufficient!

S2. No mention of m.

Insufficient!

Hence, answer will be A.
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Re: The function f is defined for all positive, three-digit integers as [#permalink] New post   21 Nov 2018, 13:45
The question doesn't mention that m and n are three digit numbers?
Is it implied?
If they are not three digit numbers then A cannot be the answer.
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Re: The function f is defined for all positive, three-digit integers as [#permalink] New post   09 Nov 2023, 06:01
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Bunuel wrote:
The function f is defined for all positive, three-digit integers as \(f(x)=(2^a)(3^b)(5^c)\), where a, b, and c are the hundreds, tens, and units digits, respectively, of x. What is m−n?
(1) \(f(m)=8f(n)\)
(2) \(n=111\)


Solution:
Pre Analysis:
  • Given for any 3-digit number x, \(f(x)=(2^a)(3^b)(5^c)\), where a, b, and c are the hundreds, tens, and units digits, respectively, of x.
  • We are asked the value of \(m-n\)

Statement 1: \(f(m)=8f(n)\)
  • Let us assume 3 digit numbers \(m=100x+10y+z\) and \(n=100p+10q+r\)
  • We need the value of \(m-n=100(x-p)+10(y-q)+(z-r)\)

  • Given \(f(m)=8f(n)\)
  • Or, \((2^x)(3^y)(5^z)=(2^3)(2^p)(3^q)(5^r)\)
  • Or, \((2^x)(3^y)(5^z)=(2^{3+p})(3^q)(5^r)\)
  • So, we can say \(x=3+p⇒x-p=3\), \(y=q⇒y-q=0\) and \(z=r⇒z-r=0\)
  • And finally \(m-n=100(x-p)+10(y-q)+(z-r)=100\times 3+10\times 0+0=300\)
  • Thus, statement 1 alone is sufficient and we can eliminate options B, C and E


Statement 2: \(n=111\)
  • No value of m is given
  • Thus, statement 2 alone is not sufficient


Hence the right answer is Option A
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Re: The function f is defined for all positive, three-digit integers as [#permalink]
09 Nov 2023, 06:01
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Bunuel
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