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 f(18) equals
(2019)
  • a)
    12
  • b)
    13
  • c)
    14
  • d)
    15
Correct answer is option 'A'. Can you explain this answer?

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Answers

Nipun Tuteja
Apr 09, 2021
Given f(mn ) = f(m) f(n) and f(24) = 54
⇒ f (24) = 2 × 3 × 3 × 3
⇒ f(2 × 12) = f(2) f(12) = f(2) f(2 × 6)
= f(2) f(2) f(6) = f(2) f(2) f(2 × 3)
= f(2) f(2) f(2) f(3) = 2 × 3 × 3 × 3
Given that f(1). f(2) and f(3) are all positive integers hence by comparison, we get
f(2) = 3 and f(3) = 2
Hear we may safely consider f(1) = 1
Now, f(18 ) = f(2) (9) = f(2) f(3 × 3)
= f(2) f(3) f(3) = 3 × 2 × 2 = 12.

Given f(mn ) = f(m) f(n) and f(24) = 54⇒ f (24) = 2 × 3 × 3 × 3⇒ f(2 × 12) = f(2) f(12) = f(2) f(2 × 6)= f(2)f(2) f(6) = f(2) f(2) f(2 × 3)= f(2) f(2) f(2) f(3) = 2 × 3 × 3 × 3Given that f(1). f(2) and f(3) are all positive integers hence by comparison, we getf(2) = 3 and f(3) = 2Hear we may safely consider f(1) = 1Now, f(18 ) = f(2) (9) = f(2) f(3 × 3)= f(2) f(3) f(3) = 3 × 2 × 2 = 12.
Given f(mn ) = f(m) f(n) and f(24) = 54⇒ f (24) = 2 × 3 × 3 × 3⇒ f(2 × 12) = f(2) f(12) = f(2) f(2 × 6)= f(2)f(2) f(6) = f(2) f(2) f(2 × 3)= f(2) f(2) f(2) f(3) = 2 × 3 × 3 × 3Given that f(1). f(2) and f(3) are all positive integers hence by comparison, we getf(2) = 3 and f(3) = 2Hear we may safely consider f(1) = 1Now, f(18 ) = f(2) (9) = f(2) f(3 × 3)= f(2) f(3) f(3) = 3 × 2 × 2 = 12.