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
a3 = (3q)3 = 27q3 = 9(3q3 ) = 9m, where m = 3q3
Case II When a = 3q + 1
In this case, we have
a3 = (3q + 1)3
⇒27q3 + 27q2 + 9q + 1
⇒9q(3q2 + 3q + 1) + 1
⇒a3 = 9m + 1, where m = q(3q2 + 3q + 1)
Case III When a = 3q + 2
In this case, we have
a3 = (3q + 1)3
⇒27q3 + 54q2 + 36q + 8
⇒9q(3q2 + 6q + 4) + 8
⇒a3 = 9m + 8, where m = q(3q2 + 6q + 4)
Hence, a3 is the form of 9m or, 9m + 1 or, 9m + 8
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