Is it constant or not?

Number Theory Level 3

Let $X$ be the set of all positive integers greater than or equal to 8 and let $f: X \to X$ be a function such that $f(x+y)=f(xy)$ for all $x$ and $y$ greater than or equal to 4. If $f(8)=9$ . Find $f(9)$ .

The answer is 9.

4 solutions

Prabhav Bansal
Aug 21, 2015

We observe that f(9)=f(20)=f(64)=f(16)=f(8)=9

Moderator note:

Nice chain of equations.

Do you think we can indeed show that this function must be the constant function?

Nice chain of equations.

Do you think we can indeed show that this function must be the constant function?

Calvin Lin Staff - 5 years, 9 months ago

I cannot say it confidently and also I have not tried yet.

Prabhav Bansal - 5 years, 9 months ago

Department 8
Aug 23, 2015

$f(9)=f(8+1)=f(8)=9$

Chew-Seong Cheong
Aug 22, 2015

$\begin{aligned} f(9) & = f(4+5) = f(20) \\ f(20) & = f(4+16) = f(64) = f(8+8) = f(16) = f(4+4) = f(8) = \boxed{9} \end{aligned}$

Jorge Jimenez
Aug 21, 2015

I've just read the problem, and now I see I didn't understand it well. What I did was the next thing: f(8)=9 f(8+1)=f(8 1), but f(8 1)=f(8) (this because 8 1=8). Hence, f(9)=f(8+1)=f(8 1)=f(8)=9. So, f(9)=9 I can assure that the function is constant

$f(1)$ does not exist.

Calvin Lin Staff - 5 years, 9 months ago

Is it constant or not?

The answer is 9.

4 solutions

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