I will provide two methods for this. Method 1 (Easy way) : USE A CALCULATOR. You will get the answer in a matter of seconds like I got. Clearly, the answer is 072. Method 2 (Slight harder way) : Here, I want to find the answer without using the calculator. Let's try it. Note :- I will be making use of Congruence Modulo and Euler's Theorem , so these are the prerequisites. Another way to put the question is “Find the remainder when 2 2017 2 2017 is divided by 1000”. First of all we factorise 1000 as: 1000 = 2 3 × 5 3 1000 = 2 3 × 5 3 Next, we find the remainder by 2 3 2 3 and 5 3 5 3 seperately. It's obvious that 2 3 2 3 divides 2 2017 2 2017 . Hence, 2 2017 ≡ 0 ( m o d 8 ) 2 2017 ≡ 0 ( m o d 8 ) Now to find the remainder by 125 (or 5^3), we use Euler's Theorem. Euler's theorem is applicable in this case since g c d ( 2 , 125 ) = 1 g c d ( 2 , 125 ) = 1 . Φ ( 125 ) = ...
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