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)?

1. 8
2. 9
3. 10
4. 11
5. 12

Correct Answer: E
Solution and Explanation:

Approach Solution 1:
This is a GMAT problem-solving question in which you have to use the details given in the question to solve the problem. The problem in this category are coming from different areas of math topics. This one particularly is from functions.
The option is given in such a way that it is difficult to guess the correct answer. The candidates need to know the right approach to get the required answer. Only one of the given five choices is correct.

Given in the question that f is a function that f(mn) = f(m)f(n) for all the positive integers. Also, it is given that f(1), f(2), and f(3) are positive integers. f(1) < f(2) and also f(24) = 54.
It has asked to find out the value of f(18)

The positive integers f (2) and f (3) are given by f (mn) = f (m) f (n) and f(1).
Since f(1) = 1 and f(4) = f(2) * f(2), we can deduce that f(2*1) = f(2)= f(2)*f(1) (2)
F(6) = f(3) * f(2) (2)
F(24) = f(4) * f(6) = f(2) * f(2) * f(2) * f(3) = 54 = 3 3 2,
that is, f(2) = 3 and f(3) = 2.
F(18)= f(9)*f(2)*f(3)*f(2)=f(3)*f(3)*f(2)=2*2*3=12

E is the correct choice.

Approach Solution 2:
This is a GMAT problem-solving question in which you have to use the details given in the question to solve the problem. The problem in this category are coming from different areas of math topics. This one particularly is from functions.
The option is given in such a way that it is difficult to guess the correct answer. The candidates need to know the right approach to get the required answer. Only one of the given five choices is correct.

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

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

E is the correct choice.

Approach Solution 3:
Let the value of f(2) and f(3) be “x” and “y” respectively

=> f(24)= f(8).f(3)= f(2).f(2).f(2).f(3)= 54
=> x^3y= 54
=> x^3y= 27*2
=> x^3y= 3^3*2
On comparing, x= 3 and y= 2

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

Therefore the value of f(18) equals 12.

“Let f be a function such that f(mn) = f(m)f(n) for every positive inte" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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