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Get Answers to all your Questions
Q5 Use Euclid’s division lemma to show that the cube of any positive integer is of the form
9m, 9m + 1 or 9m + 8.
Answers (1)
Let x be any positive integer.
It can be written in the form 3q + r where and r = 0, 1 or 2
Case 1:
For r = 0 we have
x3 = (3q)3
x3 = 27q3
x3 = 9(3q3)
x3 = 9m
Case 2:
For r = 1 we have
x3 = (3q+1)3
x3 = 27q3 + 27q2 + 9q + 1
x3 = 9(3q3 + 3q2 +q) + 1
x3 = 3m + 1
Case 3:
For r = 2 we have
x3 = (3q+2)3
x3 = 27q3 + 54q2 + 36q + 8
x3 = 9(3q3 + 6q2 +4q) + 8
x3 = 3m + 8
Hence proved.
View full answer
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