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Question 10 : 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 [TITA]
It is given that f (2) > 1 and f(mn) = f(m) f(n)
So, f (2) = f (2) f (1), as m and n are positive integers.
Only possible value for f (1) = 1
f (2) > 1
Now we know, f (24) = 54
So, f (2) f (3) f (4) = 54
f (3) f (2)3 = 54
Now we know, 54 = 2 x 33
Therefore, f (2) = 3, f (3) =2 and f (1) = 1
Now, we need to find the value of f (18)
f (18) = f (3) x f (2) x (3)
f (18) = 2 x 3 x 2 = 12
f (18) = 12
The question is "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 [TITA]"
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