## 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

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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

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Anonymous 1 year 1 Answer 233 views 2

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.