Find the last three digits of 2988^678?

• Answer is:184
  ((12*12)^4)^169*(12)^2;
  ((144)^4)^169*(144);
  (6*6)^169*(144);
  36*144;
  184;is ans;
• 11 years agoHelpfull: Yes(6) No(7)
• 144 is the answer
• 11 years agoHelpfull: Yes(4) No(4)
• http://www.youtube.com/watch?v=_ip94NlNwHo
  learn from this video because i will not be able to explain you because its complicated.
• 11 years agoHelpfull: Yes(3) No(2)
• 504 iz d ans
• 11 years agoHelpfull: Yes(1) No(0)
• This is a tough question.
  Last three digits of an expression is nothing but the remainder when we divide the given expression by 1000.
  So 29886781000=(−12)6781000=126781000
  We write 1000 as 8 ×125. So we separately find the remainders by dividing the given number by 8 and 125.
  Given expression is clearly divisible by 8. Also
  Euler totient theorem says [aϕ(N)N]Remainder=1
  Here ϕ(N) is the number of co primes of a number N. In this case ϕ(125) = 100.
  So (12100)6.1278125=16.1278125
  Now 1278=478.378=(1+15)39.(10−1)39
  (1+15)39.(−1)39(1−10)39
  - (1+39C1.15+39C2.152.... ). (1+39C1.10+39C2.102−39C3.103....
  If you observe clearly, 153 and 103 are 15 multiples. So we can omit the remaining terms.
  -(1+39.15 + 39.19.152 ).(1-39.10 + 39.19.102)
  By writing 39 as 40-1 we can simplify this equation fast.
  -(1+(40-1).15 + (40-1).19.152).(1-390 + (40-1).19.100
  -(1+100 - 15 - 19.152 ).(1-140 - 19.100)
  -(101 - 15 - (20-1).152).(1-140+100)
  -(101-15+100).(-39)
  -(61).39
  -2379 = 4
  So the remainder is 4.
  The original expression is in the format of N = 8K = 125 L + 4
  The minimum number satisfies this condition is L = 4.
  So 125 (4) + 4 = 504
  So answer is 504
• 9 years agoHelpfull: Yes(1) No(0)
• @Rohit.. How ((144)^4)^169=(6*6)^169?? Can yu please explain??
• 11 years agoHelpfull: Yes(0) No(5)
• answer is 2
• 11 years agoHelpfull: Yes(0) No(11)
• 184;is ans;
• 11 years agoHelpfull: Yes(0) No(0)