MTHE 217 - Algebraic Structures with Applications
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چکیده
(a) First realize that [8] = [3] in Z5. Take different powers of [8] until we arrive at [8] = [1]. This will help us to significantly simplify [8]. [8] = [3] = [9] = [4]; [8] = [8][8] = [8][4] = [3][4] = [2]; [8] = [8][8] = [2][3] = [1]. Hence [8] = [1]. Now 391 = (97) · 4 + 3. Thus [8] = [8](97)·4+3 = [8](97)·4[8]3 = ([8])[8] = [1][8] = [8] = [2]. Therefore the remainder when 8 is divided by 5 is 2: 8 ≡ 2 (mod 5).
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