9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 − 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
Is the above number divisible by 1 1 ?
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Let the given number be N . Then
N ≡ 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 − 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 (mod 11) ≡ 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 3 1 − 1 0 − 1 3 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 + 1 0 1 5 (mod 11) ≡ 1 0 1 5 − 1 0 (mod 11) ≡ Number of 9s = 1 4 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 + 1 0 − 1 0 (mod 11) ≡ Number of 9s = 1 4 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 (mod 11) ≡ 0 (mod 11)
Yes , N is divisible by 1 1
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( 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 0 + 1 − ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 ) m o d ( 1 1 ) =
1 − ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) m o d ( 1 1 ) = − 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 m o d ( 1 1 ) = 0