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February 4th, 2019, 12:57 AM   #1
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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.
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February 4th, 2019, 01:44 AM   #2
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Condition (b) doesn't use the wording "only when".

What exactly do you find hard to understand in relation to mathematical induction?
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February 4th, 2019, 03:18 AM   #3
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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.
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February 4th, 2019, 04:07 AM   #4
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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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