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[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 \(n\).
 one year ago
 one year 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 \(n\).
 one year ago
 one year ago

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eigenschmeigenBest ResponseYou've already chosen the best response.0
f'(a) > 0 f(a)f(b) = f(ab) f(2) = 2
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
\[f(2) = f(1 \times 2) = f(1)f(2)\] hence f(1) = 1
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
Be careful with your derivatives. This is not a continuous function.
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
i should have said x ≥ y, then f(x) ≥ f(y)
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
i was thinking induction?
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
That's what I first thought as well, but for induction to work, you need to show that \(f(3)=3\).
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
f(2^n) = 2^n doesn't particularly help
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
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...
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
thats it i think
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
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 \(f(2\cdot2)=f(2)\cdot f(2)\)
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
i did not know that
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
If it were strictly multiplicative we could say that, but unfortunately, it's not given that it's strictly multiplicative.
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
not having any luck. i tried using the fact that consecutive integers are coprime
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
let me know if you want a hint.
 one year ago

eigenschmeigenBest ResponseYou've already chosen the best response.0
i think i'll sleep on it, im too tired to do maths now lol
 one year ago

eliassaabBest ResponseYou've already chosen the best response.2
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
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
That's clever. Significantly more simple than the solution I had.
 one year ago
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