What are the last two digits of ?
Hint: Pattern
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Relevant wiki: Chinese Remainder Theorem
We need to find 2 0 1 8 2 0 1 6 m o d 1 0 0 . Since g cd ( 2 0 1 8 , 1 0 0 ) = 1 , we need to consider the factors 4 and 25 of 100 separately using Chinese remainder theorem.
Factor 4: 2 0 1 8 2 0 1 6 ≡ 0 (mod 4) .
Factor 25:
2 0 1 8 2 0 1 6 ≡ ( 2 0 0 0 + 1 8 ) 2 0 1 6 (mod 25) ≡ 1 8 2 0 1 6 (mod 25) ≡ ( 2 5 − 7 ) 2 0 1 6 (mod 25) ≡ 7 2 0 1 6 (mod 25) ≡ 4 9 1 0 0 8 (mod 25) ≡ ( 5 0 − 1 ) 1 0 0 8 (mod 25) ≡ 1 1 0 0 8 ≡ 1 (mod 25)
This implies that 2 0 1 8 2 0 1 6 ≡ 2 5 n + 1 , where n is an integer. Then we have:
2 5 n + 1 n + 1 ⟹ 2 0 1 8 2 0 1 6 ≡ 0 (mod 4) ≡ 0 (mod 4) ≡ 2 5 ( 3 ) + 1 (mod 100) ≡ 7 6 (mod 100) ⟹ n = 3