Originally Posted by

**glowplug19** If F(n) is a number-theoretic function, and therefore multiplicative (?), with

$\displaystyle F(35)=161$ and $\displaystyle F(77)=437$, find $\displaystyle F(55)$.

This was a test question, and I don't know if thats all the info. We have not gone over this much, so a good explanation of how to evaluate this would be great.

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I tried reducing everything to prime factors:

Because F(n) is multiplicative, $\displaystyle F(35)=F(5\cdot7)=F(5)\cdot F(7)=161$, where $\displaystyle 161=7\cdot23$.

Also by the same argument, $\displaystyle F(77)=F(7\cdot11)=F(7)\cdot F(11)=437$, where $\displaystyle 437=19\cdot23$.

Just looking at it, i.e. ruling out the common factor 7, I got:

$\displaystyle F(7)=23, F(5)=7, F(11)=19$ $\displaystyle \Longrightarrow F(55)=F(5\cdot11)=F(5)\cdot F(11)=(7)\cdot(19)=133$.

Is this right? I relied more on intuition than real number theory, so if it is right, how?