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Using Euclid's division lemma to show …

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Using Euclid's division lemma to show that the cube of any positive integer is of the form 9m,9m+1 or 9m+2 where m is some integer

    • Posted by Swarnima Dhabre 6 years, 10 months ago

    CBSE > Class 10 > Mathematics
    • 2 answers

    Yogita Ingle 6 years, 10 months ago

    Let a be any positive integer and b = 3
    a = 3q + r, where q ≥ 0 and 0 ≤ r < 3
     Therefore, every number can be represented as these three forms. There are three cases.
    Case 1: When a = 3q,
    Where m is an integer such that m =  (3q)3 = 27q3
    9(3q3) = 9m
    Case 2: When a = 3q + 1,
    a3 = (3q +1) 3
    a3 = 27q3 + 27q2 + 9q + 1
    a3 = 9(3q3 + 3q2 + q) + 1
    a3 = 9m + 1
    Where m is an integer such that m = (3q3 + 3q2+ q)

    Case 3: When a = 3q + 2,
    a3 = (3q +2)3
    a3 = 27q3 + 54q2 + 36q + 8
    a3 = 9(3q3 + 6q2 + 4q) + 8
    a3 = 9m + 8
    Where m is an integer such that m = (3q3 + 6q2+ 4q)
    Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.

     

    0Thank You

    Harsha Asiwal 6 years, 10 months ago

    Shayad question tha 9m,9m+1or 9m+8

    0Thank You

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