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芷晴 發問於 科學及數學數學 · 1 十年前

how to do this?? 11/11must be done!~~~~please please

Find a and b such that the table of values of f(x,y)=ax+by(mod 8)is a 8x8 Latin square. How many of the Latin squares are constructed by this way?

1 個解答

  • Ivan
    Lv 5
    1 十年前

    Fix a row in your table (i.e. fix x), then this row has all 0,1,2,,,7 if b,2b,3b...8b has all 8 different remainders (then ax+by (a,x fixed, only y change) will also cover 8 different remainder).

    From number theory this is true if and only if (b,8) is relatively prime, i.e. b=1,3,5,7. So there are 4 ways. (or you can just write out every value and check directly.)

    Similarly, fix column, we see that a=1,3,5,7.

    So the number of ways will be 4*4 = 16.

    資料來源: PhD Math
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