Q. 54.2( 421 Votes )

# Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9*m*, 9*m* + 1 or 9*m* + 8.

Answer :

Let a be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.

We know that according to Euclid's division lemma:

a = bq + r So, we have the following cases:

__Case I When a = 3q__

In this case, we have

a^{3} = (3q)^{3} = 27q^{3} = 9(3q^{3} ) = 9m, where m = 3q^{3}

__Case II When a = 3q + 1__

In this case, we have

a^{3} = (3q + 1)^{3}

⇒27q^{3} + 27q^{2} + 9q + 1

⇒9q(3q^{2} + 3q + 1) + 1

⇒a^{3} = 9m + 1, where m = q(3q^{2} + 3q + 1)

__Case III When a = 3q + 2__

In this case, we have

a^{3} = (3q + 1)^{3}

⇒27q^{3} + 54q^{2} + 36q + 8

⇒9q(3q^{2} + 6q + 4) + 8

⇒a^{3} = 9m + 8, where m = q(3q^{2} + 6q + 4)

Hence, a^{3} is the form of 9m or, 9m + 1 or, 9m + 8

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