CAT 2019 Slot 2 — QA Question 10
Basics (Functions)Easy
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
Answer & solution
Answer: 12
Solution
f(mn) = f(m) × f(n)
f(1), f(2) & f(3) are positive integers.
Now we know, f(24) = 54
So, f(2) × f(3) × f(4) = f(2) × f(3) × f(2)2 = 54
f(3) × f(2)3 = 54
Also, we know, 54 = 2 × 33
Therefore, f(2) = 3, f(3) = 2
Now, f(18) = f(3)2 × f(2)
f(18) = 22 × 3 = 12
Hence, 12.