Show that if an odd perfect number exists, it must be divisible by at least 3 different primes

Question

I would assume you'd start by showing that $\dfrac{p}{p-1}\cdot\dfrac{q}{q-1}< 2$ but I don't know how to show that nor how to continue afterwards

Answer 1

Try this with the two smallest possible odd primefactors $p_1=3,p_2=5$: then $$ \frac 32 \cdot \frac 54 = \frac {15}{8} \lt 2 $$ which is smaller than $2$ (Show, that if we take bigger primes, this value even decreases, this is easy). So this can only increase towards $2$ if we involve more primefactors...

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Additional note: just recently I've done a little self-study into this matter; perhaps this notes can be interesting/helpful for you to proceed...