what is the remainder of 18!/23?

• so according to wilson's (p-2)! / p is equals to 1
  now as 23 is prime so
  (23-2)! /23 = 1
  21! / 23 = 1
  (21*20*19)*18!/23 = 1
  (-2)*(-3)*(-4) * 18! /23 = 1
  (-24)* 18! /23 = 1
  now take the minus sign to the rhs
  so 24*18! / 23 = -1
  when 24 is divided by 23 remainder is 1 therefore
  1* 18! / 23 = -1
  
  as -1 doesnt make sense as a remainder so 23-1 = 22 is the remainder and the final answer .
• 7 years agoHelpfull: Yes(20) No(1)
• Let rem[18!/23]=r
  Wilson's Theorem says rem[(p-2)!/p]=1
  rem[21!/23]=1
  rem[21*20*19*18!/23]=1
  rem[(-2)(-3)(-4)*r/23]=1
  rem[-24r/23]=1
  24r=-23K+1 ... k=-1
  r=23+1/24
  r=1
• 8 years agoHelpfull: Yes(13) No(14)
• According to wilson theorem: { (p-1)!+1}/p=rem 0 , where p is prime no.
  so;
  (22!+1)/23= rem 0
  so;
  22!/23= rem 22 or -1
  (18!*19*20*21*22)/23= rem 22
  18!/23 *(-4)*(-3)*(-2)*(-1)= rem 22
  18!/23*24/23=rem 22
  R*1/23= rem 22
  R/23=rem 22
  hence R will be 45
• 8 years agoHelpfull: Yes(4) No(11)
• According to Wilson's Theorem
  (p-1)!=-1(modp)
  Here p is prime number(23)
  (23-1)!=-1(mod23)
  22!=-1(mod23)
  22*21*20*19*18!=-1(mod23)
  (22-23)*(21-23)(20-23)(19-23)*18!=-1(mod23)
  (-1)(-2)(-3)(-4)*18!=-1(mod23)
  24*18!=-1(mod23)
  (24-23)*18!=-1(mod23)
  18!=-1(mod23)
  Hence Remainder is -1
• 7 years agoHelpfull: Yes(4) No(1)
• Remainder is 22.
• 7 years agoHelpfull: Yes(4) No(3)
• According to Wilson Theorm
  1+(p-1)!=0(mod p) where p is a prime no
   (p-1)!= -1 (mod p)
   (23-1)!=-1(mod 23)
   22!=-1(mod 23)
   22*21*20*19*18!=-1(mod 23)
   -1*-2*-3*-4*18!=-1(mod 23) where 22/23 give -1 rem and so on
   24*18!=-1(mod 23)
   1*18!=-1(mod 23) where 24/23 give 1 rem and so on
   So rem = -1 ie 22
• 7 years agoHelpfull: Yes(3) No(0)
• remainder is 2
• 8 years agoHelpfull: Yes(1) No(3)
• so R may be -1 or 45
  
• 8 years agoHelpfull: Yes(1) No(5)
• Sachin can you explain to me this step
  
  22!/23= rem 22 or -1
  
  and
  18!/23 *(-4)*(-3)*(-2)*(-1)= rem 22
  
  how did they become -4 -3 -2 -1
• 7 years agoHelpfull: Yes(0) No(0)
• 18!=6.4023737e+15
  and 6.4023737e+15 % 23 =11
  calculate from calculator...
  
  https://www.google.co.in/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=6.4023737e%2B15+%25+23
• 7 years agoHelpfull: Yes(0) No(0)
• Ans = -1
  refer this link for detailed ans:
  https://www.google.co.in/search?q=what+is+the+remainder+of+18!%2F23%3F&oq=what+is+the+remainder+of+18!%2F23%3F&aqs=chrome..69i57.805j0j8&sourceid=chrome&ie=UTF-8
• 7 years agoHelpfull: Yes(0) No(0)
• Let rem[18!/23]=r
  Sincw, Wilson's Theorem says, rem[(p-2)!/p]=1
  rem[21!/23]=1
  rem[21*20*19*18!/23]=1
  rem[(-2)(-3)(-4)*r/23]=1
  rem[-24r/23]=1
  24r=-23K+1 ... k=-1
  r=23+1/24
  r=1
  Therefore, Answer is (a).
• 7 years agoHelpfull: Yes(0) No(1)
• remainder is 22
• 7 years agoHelpfull: Yes(0) No(0)
• As we know (p-1)!/p remainder is (-1)
  
  Now
  
  18!/23
  
  =22*21*20*19*18!/22*21*20*19*23
  
  =(23–1)!/23*22*21*20*19
  
  =(-1)/22*21*20*19
  
  = (-1)
  
  So remainder is (-1) as we know there is no means of negative remainder so remainder is 23–1=22
• 7 years agoHelpfull: Yes(0) No(1)
• A/wilson's theorem
  rem[(p-1)!/p]=p-1
  so,
  rem[22!/23]=22
  => rem[22 x 21 x 20 x19/23] x rem[18!/23]=22
  =>rem[24/23] x rem[18!/23]=22
  =>1 x rem[18!/23]=22
  hence, rem[18!/23]=22(ans)
• 7 years agoHelpfull: Yes(0) No(0)
• (p-1)!/p=p-1 if p is prime
  23 is prime
  22 will be remainder
• 6 years agoHelpfull: Yes(0) No(0)