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Using Euclid division leema. Show that …

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Using Euclid division leema. Show that the qube of any odd integer is 9q+1 or 9q+8

Posted by Aquil Ahmad 6 years, 10 months ago

CBSE > Class 10 > Mathematics

1 answers

Gaurav Seth 6 years, 10 months ago

Let a be any positive integer and b = 3
a = 3q + r, where q ≥ 0 and 0 ≤ r < 3

Therefore, every number can be represented as these three forms. There are three cases.
Case 1: When a = 3q,

Where m is an integer such that m =
Case 2: When a = 3q + 1,
a ³ = (3q +1) ³
a ³ = 27q ³ + 27q ² + 9q + 1
a ³ = 9(3q ³ + 3q ² + q) + 1
a ³ = 9m + 1
Where m is an integer such that m = (3q ³ + 3q ² + q)
Case 3: When a = 3q + 2,
a³ = (3q +2) ³
a ³ = 27q ³+ 54q ² + 36q + 8
a³= 9(3q ³+ 6q ² + 4q) + 8
a 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.

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