find the remainder when (4.4! + 5.5! + 6.6! + .. + 19.19! )/64?

• 4.4!=5!-4!
  5.5!=6!-5!
  .
  .
  19.19!=20!-19!
  
  =>solving this we get = 20!-4!
  
  20! is divisible by 64 therefore remainder is 64-24=40 ans
• 11 years agoHelpfull: Yes(44) No(5)
• remainder for 4.4!/64=32 , 5.5!/64=24 , 6.6!/64=32 , 7.7!/64=16...rest of all is divisible by 64 so net remainder will be (32+24+32+16)/64 = 40..
  
  so answer will be 40...
• 11 years agoHelpfull: Yes(15) No(1)
• 4*4!= rem (1/(2^6))=1 it contain 2^5 in num thus rem(15/2)=1
  5*5!= rem 3 it contain 2^3 in num thus rem(75/8)=3
  6*6!= rem 1 it contain 2^5 in num thus rem(27/2)=1
  7*7!= rem 1 it contain 2^5 in num tus rem(7*7*3*5*3/2)=1
  8*8! and above are divisible by 64
  thus ans 6
• 11 years agoHelpfull: Yes(9) No(9)
• Notice first that 8! is divisible by 64, as 8 = 1.2.3.4.5.6.7.8 and 2.4.8 = 64;
  we write 8! | 64
  
  By the same argument j! | 64 for all j >= 8. Thus in the sum of the question,
  S = 1.1! + 2.2! + ... + 19.19!
  we can disregard all terms after 7.7! as all of them will have remainder 0 when divided by 64.
  Now term by term,
  4.4! = 96, r = 32
  5.5! = 600, r = 24
  6.6! = 4320, r = 32
  7.7! = 35,280, r = 16
  The sum is 104 which has remainder 40
  So simply 40 is the answer!!
• 9 years agoHelpfull: Yes(4) No(0)
• so the solution as follow this question is totaly depend upon only one thing that is 5! jus see 5!=120 6!=6*120 7!=7*6*120 here we notice that after 5! factorial we get 0 because when 0 is multiply with any no. it get 0 ...whether it is 6!,7!,8!,9!,10!,11! and so on, all have zero at the end .. so notice to question 4.4!+5.120+xx0+xx0+xx0....xx will be any no. , i did nt write because we have to focus the zero which is at the end only ...4.4!+xx0=4.4.3.2.1+xx0=96+xx0=xx6......so we get 6 at the end...is it seen??
  yes you are so...xx6/64..... now jus check 4 table ( 4 8 12 16 20 24 28 32 36 40) xx6 is divisible by 64 only at the 4*4=16 (6 at the end) 4*9 =36 (6 at the end) 64 devide xx6 no at 4 times or 9 times ...at question we have only one option who has last digit is 4..so that one is answer..i didnt remember the option..hope you understand it.
• 11 years agoHelpfull: Yes(1) No(2)
• 1+3+1+1=6 is the ans....
  
  
  
  
• 11 years agoHelpfull: Yes(0) No(5)