(6^1001+7^1001)/17
find the reminder

• 6^1001/17
  =(6^2)^500 *6/17
  = (17*2+2)^500 *6/17
  = 2^500 *6/17
  = (2^4)^125 *6/17
  = (17-1)^125 *6/17
  = -6/17
  
  7^1001/17
  =(7^2)^500 *7/17
  =(17*3-2)^500 *7/17
  = (-2)^500 *7/17
  = (2^4)^125 *7/17
  = (17-1)^125 *7/17
  = -7/17
  
  so, (6^1001+7^1001)/17 => rem(-6/17 + (-7/17))=> rem(-13/17) => rem(4/17)
  
  remainder is 4
• 10 years agoHelpfull: Yes(25) No(1)
• 6*6=36 36/17 remainder 2
  7*7 =49 49/17 remainder 15
  
  (6^2)^500 * 6 + (7^2)^500 * 7
  
  remainder 2^500 * 6 + 15^500 *7
  2^5=32 32/17=15 15^2=225 225/17=4 both remainder
  (2^5)^100 *6 + (15^2)^250 *7
  15^100 *6 + 4^250 *7
  4^50 * 6 + 13^81 *28
  13^16 *96 + 16^40 *364
  16^8 *96 + 1^20 *364
  1^4 *96 +1^20 *364
  96+364=460
  460/17=27 remainder is 1.
• 10 years agoHelpfull: Yes(6) No(4)
• 6^1001 + 7^1001
  (6^2)^500*6 + (-10)^1001
  (2^4)^125*6 + 100^500*(-10)
  
  (-1)^125*6 + (-2)^500*(-10)
  -6 + (-2^4)^125*(-10)
  -6 + (-1)(-10)
  -6 + 10
  4 remainder = answer.
• 10 years agoHelpfull: Yes(3) No(0)
• according to-->
  6/7 has remainder 6
  36/7 has remainder 1
  216/7 has remainder 6
  1296/7 has remainder 1
  
  We see that 6 to an odd power divided by 7 has remainder 6
  and 6 to an even power divided by 7 has remainder 1
  
  so 6^1001 mod 7 being an odd power will have a remainder of 6.
  
  We could say that 6 in mod 7 is -1
  so -1^1001 mod7 = -1 mod 7....and since 7-1=6 the answer is 6.
  
  now let go to our example
  
  (6^1001+7^1001)/7
  = 6^1001/7=6 and 7^1001/7=0
  =6+0=1 ans
• 10 years agoHelpfull: Yes(1) No(14)
• 4 remainder will the answer...
• 10 years agoHelpfull: Yes(1) No(0)
• 6^1+7^1=13/17 so remainder 4
• 9 years agoHelpfull: Yes(1) No(0)
• can anyone expln in shrtct mthd plzzzzzzz
  
• 10 years agoHelpfull: Yes(0) No(0)
• remainder 1
• 5 years agoHelpfull: Yes(0) No(0)