Funny way to pull the -1 out from the parenthesis.
p * (-q) = p * (-1) * q = p * q * (-1) (mod pq)
That is, p * (-q) = 0 (mod pq).
Well, let's proof:
some days ago RSA-640 was factored, therefore I'll use this number for proofing.
N = p*q =
3107418240490043721350750035888567930037346022842727545720161948823206440518081504556346829671723286782437916272838033415471073108501919548529007337724822783525742386454014691736602477652346609
p =
1634733645809253848443133883865090859841783670033092312181110852389333100104508151212118167511579
q =
1900871281664822113126851573935413975471896789968515493666638539088027103802104498957191261465571
Hence p*(-q) = p*(N-q), we have:
1634733645809253848443133883865090859841783670033092312181110852389333100104508151212118167511579*(3107418240490043721350750035888567930037346022842727545720161948823206440518081504556346829671723286782437916272838033415471073108501919548529007337724822783525742386454014691736602477652346609-1900871281664822113126851573935413975471896789968515493666638539088027103802104498957191261465571)
=
5079801149330465928652035530544913704964519649664113022948507643221268839586387905945718488562426349551024378408981587404238854112680081565808050803367178098655476230508056302202082021498932996241380749611265431048278537997959344921052965979997472486960464297533557254211807262177876539002;
and, by my gypothesis:
p*(-q) = p*q *(p-1) = p*(N-q)
163473364580925384844313388386509085984178363092312181110852389333100104508151212118167511579*1900871281664822113126851573935413975471896789968515493666638539088027103802104498957191261465571*1634733645809253848443133883865090859841783670033092312181110852389333100104508151212118167511578
=
5079801149330465928652035530544913704964519649664113022948507643221268839586387905945718488562426349551024378408981587404238854112680081565808050803367178098655476230508056302202082021498932996241380749611265431048278537997959344921052965979997472486960464297533557254211807262177876539002;
Q.E.D
Any new idea?
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