Let f be a function such that f(mn) = f(m)f(n) for every positive inte

Question

Let f be a function such that f(mn) = f(m)f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then what is the value of f(18)?

A. 8
B. 9
C. 10
D. 11
E. 12

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GMATinsight · Accepted Answer

Given f (mn) = f (m) f (n)
and 
f(1), f (2) and f (3) are positive integers.

As we know f(2*1) = f(2)= f(2)*f(1),  so f(1) = 1 
f(4) = f(2) * f(2)
F(6) = f(3) * f(2)
F(24) = f(4) * f(6) = f(2)* f(2) * f(2) * f(3)  = 54 = 3^3 *2 ,

i.e. f(2) = 3 and f(3) =2

F(18) = f(9) * f(2) = f(3) * f(3) * f(2) = 2*2*3 = 12

Answer: Option E

BrentGMATPrepNow · Answer

Given: f(24) = 54
Since f(mn) = f(m)f(n), we can write:      = 54
Find prime factorization of 54 to get:          =  (3)(3)(3)  (2) 
From this we can conclude that:  f(2) = 3  and  f(3) = 2

So.... f(18) =        
=  (3)  (2)(2) 
= 12

Answer: E

Cheers,
Brent

Kinshook · Answer

Let f be a function such that f(mn) = f(m)f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then what is the value of f(18)?

f(1) = f(1)*f(1) ; f(1) = 1
f(2) > 1; Let f(2) be a; and let f(3) = b; where a & b are positive integers
f(24) = {f(2)}^3*f(3) = 54
a^3*b = 54 = 3^3*2; a = 3; b = 2
f(18) = {f(3)}^2*f(2) = b^2*a = 4*3 = 12

IMO E

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=HVxTk4euN5c

f(mn) = f(m)f(n)

f(24)  = f(2*12) = f(2) * f(12) = f(2) * f(4*3) = f(2) * f(4) * f(3)
= f(2) * f(2*2) * f(3) = f(2) * f(2) * f(2) * f(3)=  ((f(2))^3 * f(3)  = 54 =  3^3 * 2 
=> f(2) = 3 and f(3) = 2

f(18)  = f(9*2) = f(9) * f(2) = f(3*3) * f(2) = f(3) * f(3) * f(2) = 2 * 2 * 3 = 12

So,  Answer will be E 
Hope it helps!

Watch the following video to learn the Basics of Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

bumpbot · Answer

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Let f be a function such that f(mn) = f(m)f(n) for every positive inte

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