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

Question

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

14101992 · Accepted Answer

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.

benignfiend · Answer

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.

SaquibHGMATWhiz · Answer

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	imes 3+10	imes 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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The function f is defined for all positive, three-digit integers as

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