Use Euclid’s use equally division lemma to show that cube of any positive integer is of the form of 9M ,9M + 1 or 9M + 8

Question

Use Euclid’s use equally division lemma to show that cube of any positive integer is of the form of 9M ,9M + 1 or 9M + 8

solved 0
Anonymous 1 year 1 Answer 233 views 2

Answer ( 1 )

  1. As per Euclid’s Division Lemma

    If a and b are two positive integers, then

    a = bq + r , where 0 ≤ r < b

    Let a be any positive integer and b = 3

    Hence, a = 3q + r, where q ≥ 0 and 0 ≤ r < 3

    Therefore, there are three cases.

    Case 1: If r = 0

    Our equation becomes

    a = 3q

    Cubing both sides

    a3 =  (3q)3 = 27q3
    a3 = 9(3q3) = 9m

    Where m is an integer such that m = 3q3

    Case 2: If  r = 1

    Our equation becomes

    a = 3q + 1

    Cubing both sides

    3 = (3q +1) 3
    3 = 27q 3 + 27q 2 + 9q + 1
    3 = 9(3q 3 + 3q 2 + q) + 1
    3 = 9m + 1

    Where m is an integer such that m = (3q 3 + 3q 2+ q)

    Case 3:  r = 2

    Our equation becomes

    a = 3q + 2

    Cubing both sides

    a 3 = (3q +2) 3
    3 = 27q 3 + 54q 2 + 36q + 8
    3 = 9(3q 3 + 6q 2 + 4q) + 8
    3 = 9m + 8

    Where m is an integer such that m = (3q 3 + 6q 2+ 4q)

    Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.

     

    Best answer

Leave an answer