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Question 31 : If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is [TITA]

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Given that f(ab) = f(a) × f(b) for all possible integers a and b.

We have to find the largest possible value of f(1). We know that,

f(ab) = f(a) × f(b)

⟹ f(a × 1) = f(a) × f(1)

⟹ [f(a) × f(1)] – f(a) = 0

⟹ f(a)[f(1) – 1] = 0

There are two possibilities here, either f(a) = 0 or f(1) – 1 = 0.

If f(a) = 0, then it is a constant function. Hence, f(1) = 0.

But, if f(1) – 1 = 0, f(1) = 1.

Hence, f(1) = 0 or f(1) = 1.

Out of these, f(1) = 1 is the largest possible value.

The question is **"If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is [TITA]" **

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