Remainder when 97! is divided by 101????

• 97!/101
  
  1*2*3*4....*97/101
  
  rem(1/101)+rem(2/101)+rem(3/101)....+rem(97/101)
  
  rem((1+2+3+.....+97)/101)
  
  rem(4753/101)
  
  =6(ans)
• 10 years agoHelpfull: Yes(27) No(35)
• according to wilsons theorem...(n-1)!=-1(mod n)
  here n is 101
  so 100!=-1(mod 101)
  100*99*98*97!=-1(mod 101)
  97!={-1*(100*99*98)}(mod 101)
  97!={-1*-1*-2*-3} (mod 101)
  97!=6 (mod 101)
  so answer is 6
  
• 10 years agoHelpfull: Yes(11) No(0)
• 17....using wilson's theorem
• 10 years agoHelpfull: Yes(6) No(2)
• Wilson's Theorem is about primes, namely, for integer n>1,
  (n-1)! = -1 mod n iff n is prime.
  
  Since 101 is prime, we can use it, along with the fact that
  100*99*98*97! = 100!
  
  Then taking mod 101 and using Wilson's Theorem,
  100*99*98*97! = -1 mod 101
  (-1)(-2)(-3)97! = -1 mod 101
  
  so that, multiplying both sides by -1,
  6*97! = 1 mod 101
  
  Now, we'd like to find the inverse mod 101 of 6. To do this, use the fact that
  6*17 = 102 = 1 mod 101, to multiply both sides by 17, yielding
  
  17*6*97! = 97!mod 101 = 17 mod 101
  
  So it's 17.
• 10 years agoHelpfull: Yes(4) No(0)
• Wilsom theorem states:
  (n-1)!=-1 mod n if n is prime
  Since, 101 is luckily prime
  100!=-1 mod 101
  now, 100*99*98*97!=-1 mod 101
  inverse of 100=100 or (-1), of 99=50, of 98 is 67
  so 97!=-1*100*50*67 mod 101
  or 97!=-1*-1*50*67 mod 101
  =3350 mod 101
  =17 (ans)
• 10 years agoHelpfull: Yes(3) No(1)
• 97!/101=
  97*96*............1/101
  (97/101)+(96/101).......1/101
  (1+2+3+4..........96+97)/101
  (((1+97)/2)*97)/101=6
• 9 years agoHelpfull: Yes(2) No(0)
• Since 101 is prime, any number in range 1 to 100 is invertible
  Use Extended Euclidean algorithm given here http://en.wikipedia.org/wiki/Modular_multiplicative_inverse
  A tabular method can be taken from any book on modular arithmetic.
• 10 years agoHelpfull: Yes(1) No(0)
• how inverse of 99 and 98 are 50 and 67 respectivly? pls explain
• 10 years agoHelpfull: Yes(1) No(0)
• As already said, it is calculated using Extended Eucledian theorem.
  It means 99*50 mod 101=1 98*67 mod 101=1
• 10 years agoHelpfull: Yes(1) No(0)
• @D G please explain ur answer
• 10 years agoHelpfull: Yes(0) No(0)
• @ D G. wats wilson theorem?
• 10 years agoHelpfull: Yes(0) No(0)
• @ D G. wats wilson theorem?
• 10 years agoHelpfull: Yes(0) No(0)
• pranshu plzzzz tel me that step .... 97! vala jo first step hai,,why did u take 97!,,??????????
• 10 years agoHelpfull: Yes(0) No(0)
• @MD...bcoz in dis prblm v hv to calculate 97%101
• 10 years agoHelpfull: Yes(0) No(0)
• @prateek..please explain the concept of taking inverse which u hv taken..
• 10 years agoHelpfull: Yes(0) No(1)
• remainder 97 and quotient is 0..
• 9 years agoHelpfull: Yes(0) No(0)