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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]
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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