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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]

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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 3^{3}

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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