Yield beta and conversion factor's place in formulas calculating hedge ratio and # of futures

I am clear with the number of contract calculation

Number of contracts = {[(target duration) - (portfolio duration)]/(futures duration)} * {(Value of portfolio)/(value of futures contract * multiplier)}

This makes sense. You need fewer futures contracts if they have high multiplier

Fixed Income - duration management

Number of contracts = (same ratio of durations as above) * {(Value of portfolio * CTD conversion factor)/(value of CTD)}

This also make sense, the value fo CTD should divide by conversion factor

BUUUUUT

Hedge ratio=DH*PH/ (D-CTD*P-CTD) ×Conversion factor for the CTD bond×Yield beta

shouldn’t it divide the conversion factor for the CTD bond instead of multiply? Maybe I get the idea of hedge ratio wrong