It is easier than you think!
What is the remainder if 2 7 8 1 5 3 3 5 is divided by 2 0 ?
The answer is 1.
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2 solutions
From the identity a n − b n = ( a − b ) ( a n − 1 + b a n − 2 + . . . b n − 1 ) we know that 2 7 8 1 5 3 3 5 − 1 = ( 2 7 8 1 − 1 ) ( 2 7 8 1 5 3 3 5 − 1 + 2 7 8 1 5 3 3 5 − 2 + 1 1 − 1 ) .
Because 2 7 8 1 − 1 is divisible by 2 0 , the left-hand side ( 2 7 8 1 5 3 3 5 − 1 is divisible by 2 0 so 2 0 2 7 8 1 5 3 3 5 will give us a remainder of 1.
*Most of the solution by Ishtiaque Ahmed Sayef.
As you have edited your comment, I have also updated my comment.
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Selam Aleykum, thanks for bringing this up. I've changed it - does it seem better now.
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Wa alaikumussalam little brother. I am very much happy that you wished me salam. The pattern you are talking about in the edited comment is correct and I have understood what you have tried to express. But your solution procedure is not well organized. If you don't want to use modular arithmetic, then you can use the fact that a n − b n = ( a − b ) ( a n − 1 + a n − 2 b + . . . + b n − 1 ) , that is 2 7 8 1 5 3 3 5 − 1 = ( 2 7 8 1 − 1 ) ( 2 7 8 1 5 3 3 5 − 1 + 2 7 8 1 5 3 3 5 − 2 + . . . + 1 ) . Since ( 2 7 8 1 − 1 ) is divisible by 2 0 , the left hand side 2 7 8 1 5 3 3 5 − 1 is also divisible by 2 0 . Thus if 2 7 8 1 5 3 3 5 is divided by 2 0 , 1 is the remainder.
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@Ishtiaque Ahmed Sayef – Thanks! I hadn't heard of this formula a n − b n formula before.
2 7 8 1 5 3 3 5 ≡ ( 2 7 8 0 + 1 ) 5 3 3 5 ≡ 1 5 3 3 5 ≡ 1 (mod 20)