What is the second least signifficant digit of ?
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Let A = ( 1 0 + 9 0 ) k where k is the exponent given in the problem. The only important properties of k are that it is big and it is even. We may write A = ∑ i = 0 k ( i k ) 9 0 i 1 0 k − i . Let's also define B = ( 1 0 − 9 0 ) k = ∑ i = 0 k ( i k ) ( − 1 ) i 9 0 i 1 0 k − i . Notice that A + B = 2 ∑ i = 0 2 k ( 2 i k ) 9 0 2 i 1 0 k − 2 i = 2 ∑ i = 0 2 k ( 2 i k ) 9 0 i 1 0 k − 2 i . What is important about this number is that it is an integer. And thus ⌊ A ⌋ = ⌊ A + B − B ⌋ = A + B + ⌊ − B ⌋ . Because B is a positive number less than 1 we already know that ⌊ − B ⌋ = − 1 so the previous expression is just A + B − 1 .
To find the second digit just compute A + B − 1 ≡ − 1 ≡ 9 9 m o d 1 0 0 . The answer is 9.