CBSE Class 10 Answered
use euclid's division lemma to show that the cube of any positive integer is either of the form 9m,9m+1,9m+8 for some integer m.
Asked by | 14 Apr, 2012, 12:30: PM
Expert Answer
Let x be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.
So, we have the following cases :
Case I : When x = 3q.
then, x3 = (3q)3 = 27q3 = 9 (3q3) = 9m, where m = 3q3.
Case II : When x = 3q + 1
then, x3 = (3q + 1)3
= 27q3 + 27q2 + 9q + 1
= 9 q (3q2 + 3q + 1) + 1
= 9m + 1, where m = q (3q2 + 3q + 1)
Case III. When x = 3q + 2
then, x3 = (3q + 2)3
= 27 q3 + 54q2 + 36q + 8
= 9q (3q2 + 6q + 4) + 8
= 9 m + 8, where m = q (3q2 + 6q + 4)
Hence, x3 is either of the form 9 m or 9 m + 1 or, 9 m + 8.
Answered by | 08 May, 2012, 10:53: AM
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