N=1991*1993*1995*1997*1999*2001*2003*2005*2007 What is the sum of the hundreds, tens and units digits of N?
N=1991*1993*1995*1997*1999*2001*2003*2005*2007, solve for N
N = 509685296446883641136997950775
7 + 7 + 5 ==19
+118608 Now without a calculator:
1991*1993*1995*1997*1999*2001*2003*2005*2007 (only care about last 3 digits)
same as
(2000-7)(2000+7)(2000-5)(2000+5)(2000-3)(2000+3)(2000-1)(2000+1)(2000-9) mod1000
= (1000-49)(1000-25)(1000-9)(1000-1)(1000-9) mod 1000
= (-49*-25*-9*-1*-9) Mod1000
= -(49*25*81) mod1000
= -(49*25*80+49*25) mod1000
= -(49*2000+49*25) mod1000
= -(49*25) mod1000
= -(40*25+9*25) mod1000
= -(9*25) mod 1000
= -225 mod 1000
= 1000-225 mod1000
= 775 mod 1000
7+7+5 = 19
