## KingGeorge 3 years ago [SOLVED by @eliassaab] Define a function $$f: \mathbb{Z}^+ \longrightarrow \mathbb{Z}^+$$ such that $$f$$ is strictly increasing, $$f$$ is multiplicative, and $$f(2)=2$$. Show that $$f(n)=n$$ for all $$n$$.

1. eigenschmeigen

f'(a) > 0 f(a)f(b) = f(ab) f(2) = 2

2. eigenschmeigen

$f(2) = f(1 \times 2) = f(1)f(2)$ hence f(1) = 1

3. KingGeorge

Be careful with your derivatives. This is not a continuous function.

4. eigenschmeigen

ah no it isnt

5. eigenschmeigen

i should have said x ≥ y, then f(x) ≥ f(y)

6. eigenschmeigen

i was thinking induction?

7. KingGeorge

That's what I first thought as well, but for induction to work, you need to show that $$f(3)=3$$.

8. eigenschmeigen

f(2^n) = 2^n doesn't particularly help

9. eigenschmeigen

ah it does, now we know f(4) cant we just say $f(2) < f(3) < f(4)$ and since f(3) is an integer...

10. eigenschmeigen

thats it i think

11. KingGeorge

Multiplicative means that if $$\gcd(a, b)=1$$, then $$f(a\cdot b)=f(a)\cdot f(b)$$ $$\gcd(2, 2)=2$$ so we can't say that $$f(2\cdot2)=f(2)\cdot f(2)$$

12. eigenschmeigen

ah

13. eigenschmeigen

i did not know that

14. KingGeorge

If it were strictly multiplicative we could say that, but unfortunately, it's not given that it's strictly multiplicative.

15. eigenschmeigen

not having any luck. i tried using the fact that consecutive integers are coprime

16. KingGeorge

let me know if you want a hint.

17. eigenschmeigen

i think i'll sleep on it, im too tired to do maths now lol

18. eliassaab

f(3)f(5)=f(15)<f(18) =f(2) f(9)=2f(9)< 2 f(10) =4 f(5) Henc2 f(2)=2<f(3) < 4 f(3) =3

19. KingGeorge

That's clever. Significantly more simple than the solution I had.