what is the remainder when 7^105 is divided by 25

• 7^1 % 25 = 7
  7^2 % 25 = 24
  7^3 % 25 = 18
  7^4 % 25 = 1
  7^5 % 25 = 7
  7^6 % 25 = 24
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  So it continues like 7, 24, 18, 1, 7, 24... (4 cycles)
  
  105/4= 26 * 4 + 1 [Since v have 4 cycles, v divide it by 4]
  
  it completes 26 cycles of {7, 24, 18 ,1} and again continues from 7
  
  So the remainder is 7
  
• 10 years agoHelpfull: Yes(18) No(0)
• 7^0/25= 0 with remainder 1
  7^1/25= 0 with remainder 7
  7^2/25= 1 with remainder 25
  7^3/25= 13 with remainder 18
  7^4/25= 96 with remainder 1(repeating pattern)
  
  Hence every 4th exponent, pattern is repeating.
  105/4 = 26.25
  Take only integer part and multiply by 4=> 26*4= 104
  Subtract it from 105 to get 1
  
  Hence, answer is same remainder as that of 7^1 ie 7
  
  (MAYBE)
  
• 10 years agoHelpfull: Yes(1) No(0)