February 4th, 2019, 12:57 AM  #1 
Newbie Joined: Feb 2019 From: home Posts: 5 Thanks: 0  mathematics help Problem 2.2 Find all functions $f$ : $\mathbb{N}\to \mathbb{N}$ which satisfy (a) $f(2) = 2$; (b) $f(mn) = f(m)f(n)$ for all $m,n$ in $\mathbb{N}$ satisfying the condition $\gcd(m,n) = 1$; (c) $f(m) < f(n)$ whenever $m < n$. This is a question from my book. I attempted it as and The book uses induction to solve (though I couldn't understood that). Have I correctly attempted it? And is it possible to include induction anywhere? Thank you. Last edited by skipjack; February 4th, 2019 at 01:31 AM. 
February 4th, 2019, 01:44 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,262 Thanks: 1958 
Condition (b) doesn't use the wording "only when". What exactly do you find hard to understand in relation to mathematical induction? 
February 4th, 2019, 03:18 AM  #3 
Newbie Joined: Feb 2019 From: home Posts: 5 Thanks: 0 
Condition (b) in question implicitly telling 'only when'. it writes: f(mn)=f(n)f(m) where m,n satisfying the condition gcd(m,n)=1 Suppose I want to check my answer through induction, then how do I? Can you provide steps? Last edited by skipjack; February 4th, 2019 at 03:58 AM. 
February 4th, 2019, 04:07 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,262 Thanks: 1958 
The problem's wording isn't intended to mean "only when". The problem leaves it to you to determine whether $f(mn) = f(m)f(n)$ is also true when the given condition isn't satisfied. Mathematical induction is a proof technique. 

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