Think Logically not mathematically [part-32]

Algebra Level 3

{ f ( m n ) = f ( m ) f ( n ) f ( m ) < f ( n ) if m < n f ( 2 ) = 2 \begin{cases} f(mn) & = & f(m)f(n) \\ f(m) & < & f(n) & \text{if} \ \ m<n \\ f(2)& = & 2 \end{cases}

Suppose f : N → N f: \mathbb {N \rightarrow N} is a function satisfying the conditions above. Find the value of ∑ k = 1 20 f ( k ) \displaystyle \sum_{k=1}^{20} f(k) .


The answer is 210.

Sudoku Subbu by sudoku subbu , 19, India

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1 solution

Sudoku Subbu
Jun 26, 2016

Given, f ( m n ) = f ( m ) f ( n ) f(mn)=f(m)f(n) and also f ( 2 ) = 2 f(2)=2 = > f ( 2 × 1 ) = f ( 2 ) f ( 1 ) => f(2\times1)=f(2)f(1) = > f ( 1 ) = 1 => f(1)=1 f ( 4 ) = f ( 2 ) f ( 2 ) = 2 × 2 = 4 f(4)=f(2)f(2)=2\times2=4 w e k n o w f ( 4 ) = 4 a n d f ( 2 ) = 2 we \space know f(4)=4 \space \space and f(2)=2 Given , f(m)>f(n) if m>n = > f ( 3 ) = 3 => f(3)=3 from this we get that f(k)=k T h e r e f o r e ∑ k = 1 20 = 1 + 2 + 3 + ⋯ + 20 = 210 Therefore \space \displaystyle \sum_{k=1}^{20}=1+2+3+ \dots +20=210

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