what is the last two digits of 259^66

• The given problem can be expressed as
  
  (250 + 9) ^ 66 = 250^ 66( that is 66 C 1) + 66 * 250^ 65 * 9 + .............
  + 66* 250 * 9 ^ 65 + 9 ^ 66
  
  All the first 65 terms have 250 raised to at least a power of 2, so all these terms have the last two digits as zeroes
  The 66 th term has 66 * 250, so the last two digits again will be zero.
  
  So, the only term that influences the answer is the last term, 9^ 66
  
  So, we have to simply understand the pattern of the last two digits of 9 powers
  
  The pattern for 9 power 1, 2, 3, 4 ,5 and so on is
  
  09, 81, 29, 61 we can observe an alternate series of increase and decrease of 20
  
  So, the first 20 terms are
  09, 81, 29, 61, 49, 41, 69, 21, 89, 01, The series again continues 09, 81, 29, 61
  
  
  So, the given expression is following a cycle of 10. So, 66 th term will be same as 6 th term, 41 will be the answer.
  
  Answer is 41.
  
• 10 years agoHelpfull: Yes(9) No(1)
• 259^66=67081^33
  last two digit is 81.
  so 81^33= 531441^11
  last two digit is 41.
  now....
  41^2=1681
  41^3=68921
  41^4=2825761
  41^5=115856201
  
  means last two digit is 01
  
  so we can write...
  41^11=41^5 * 41^5 * 41
  =01*01*41
  =41.
  so 41 WILL BE ANSWER ..
• 10 years agoHelpfull: Yes(4) No(0)
• (259)^66=(259^2)^33=(last two digit of 259^2)^33=(91)^33=last two digit of (91)^33=71
• 10 years agoHelpfull: Yes(3) No(5)
• (259)^66=(259^2)^33=(last two digit of 259^2)^33=(91)^33=last two digit of (81)^33=41
• 10 years agoHelpfull: Yes(2) No(0)
• ans will be 29
• 10 years agoHelpfull: Yes(1) No(3)
• we have to calculate last to digit
  (259)^66
  means 59^66=(59^2)^33=(81)^33
  i.e last digit 1
  second last 8*3=24 means 4
  so 41 is the ans
• 9 years agoHelpfull: Yes(1) No(0)
• 9^2n=(.....1).. so it will be 1!
• 10 years agoHelpfull: Yes(0) No(3)
• simply ((259)^2)^33)= (67081)^33 >>>> last two digit will be 41
• 9 years agoHelpfull: Yes(0) No(0)
• last digit=9^66
  =(3^2)^66
  =(3^4)^33
  =81^33
  1^3=1
  8*3=24
  then last two digit=41
• 7 years agoHelpfull: Yes(0) No(0)