Let f be a function of positive integers which takes integer values and have the following properties.
Find f ( 1 9 8 3 ) .
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Nice solution buddy (+1)
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@Vicky Vignesh are you interested brother?
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mhm.Nope. Please don't mention me in a problem I've not solved.
Isnt it an identity function?
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What is it?
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The value we put the value we get like g can be an identity function if g (x)=x
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@Md Zuhair – So , you mean to say that f ( x ) is an identical function if f ( x ) = x , If that is the definition , then it is clearly an identity function.
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f ( 2 n ) = f ( 2 n − 1 ) ⋅ f ( 2 ) = f ( 2 n − 2 ) ⋅ f ( 2 ) ⋅ f ( 2 ) = . . . . . . . = ( f ( 2 ) ) n = 2 n
⇒ f ( 2 a ) = 2 a , f ( 2 a + 1 ) = 2 a + 1
Now , there are 2 a − 1 integers between 2 a and 2 a + 1 , so the function can be applied on 2 a − 1 values and it corresponds to exactly 2 a − 1 values and since the function is increasing , therefore for all values of x , f ( x ) = x ∀ x ∈ ( 2 a , 2 a + 1 , ⋯ , 2 a + 1 )
Therefore ∀ x ∈ N , f ( x ) = x
And hence f ( 1 9 8 3 ) = 1 9 8 3