Problem on Function

Algebra Level 3

Let f f be a function of positive integers which takes integer values and have the following properties.

  1. f ( 2 ) = 2 f(2) = 2
  2. f ( m n ) = f ( m ) f ( n ) f(mn) = f(m) f(n)
  3. f ( m ) > f ( n ) f(m) > f(n) , if m > n m>n

Find f ( 1983 ) f(1983) .


The answer is 1983.

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1 solution

Ankit Kumar Jain
Apr 8, 2017

f ( 2 n ) = f ( 2 n 1 ) f ( 2 ) = f ( 2 n 2 ) f ( 2 ) f ( 2 ) = . . . . . . . = ( f ( 2 ) ) n = 2 n f(2^{n}) = f(2^{n - 1})\cdot f(2) = f(2^{n - 2})\cdot f(2)\cdot f(2) = .......= (f(2))^{n} = 2^{n}

f ( 2 a ) = 2 a , f ( 2 a + 1 ) = 2 a + 1 \Rightarrow f(2^{a}) = 2^{a} , f(2^{a+1}) = 2^{a+1}

Now , there are 2 a 1 2^{a} - 1 integers between 2 a 2^a and 2 a + 1 2^{a+1} , so the function can be applied on 2 a 1 2^a - 1 values and it corresponds to exactly 2 a 1 2^a - 1 values and since the function is increasing , therefore for all values of x x , f ( x ) = x f(x) = x x ( 2 a , 2 a + 1 , , 2 a + 1 ) \forall x \in (2^a , 2^a + 1 , \cdots , 2^{a+1})

Therefore x N , f ( x ) = x \forall x \in \mathbb N , f(x) = x

And hence f ( 1983 ) = 1983 f(1983) = 1983

Nice solution buddy (+1)

Rahil Sehgal - 4 years, 2 months ago

Thanks!

I am making an edit to the solution.

Ankit Kumar Jain - 4 years, 2 months ago

@Satwik Murarka and @Saswata Naha . I, @Rahil Sehgal and @Ankit Kumar Jain have made an whatsapp group for Brilliant discussions and problem solving games. Are you two interested to join? If so please comment your phone number.

ThankYou

Md Zuhair - 4 years, 2 months ago

@Vicky Vignesh . I, @Rahil Sehgal and @Ankit Kumar Jain have made an whatsapp group for Brilliant discussions and problem solving games. Are you two interested to join? If so please comment your phone number

Md Zuhair - 4 years, 2 months ago

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@Vicky Vignesh are you interested brother?

Md Zuhair - 4 years, 2 months ago

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mhm.Nope. Please don't mention me in a problem I've not solved.

Viki Zeta - 4 years, 2 months ago

Isnt it an identity function?

Md Zuhair - 4 years, 2 months ago

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What is it?

Ankit Kumar Jain - 4 years, 2 months ago

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The value we put the value we get like g can be an identity function if g (x)=x

Md Zuhair - 4 years, 2 months ago

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@Md Zuhair So , you mean to say that f ( x ) f(x) is an identical function if f ( x ) = x f(x) = x , If that is the definition , then it is clearly an identity function.

Ankit Kumar Jain - 4 years, 2 months ago

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@Ankit Kumar Jain Yes. that I wanted to say

Md Zuhair - 4 years, 2 months ago

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