Question

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

Answer 1

answer of q no 5 of ex 1.1

Answer 2

Hi there !!

Let "a" be any positive integer and
b = 3

BY Euclid's division lemma ;

a = 3q + r ,
where r = 0 , 1 , 2

When r = 0

a = 3q

a³ = (3q)³
= 27q³
= 9(3q³)
= 9 m , where m = 3q³
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WHen r = 1

a = 3q + 1

a³ = (3q + 1)³
= 27q³ + 27q² +9q + 1
= 9(3q³ + 9q² + 3q ) + 1
= 9 m + 1 , where m = 3q³ + 9q² + 3q
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When r = 2,
a = 3q + 2

a³ = (3q + 2)³
= 27 q³ + 54q² + 36q + 8
= 9 ( 3q³ +6q² + 4q)+8
= 9 m + 8 , where m = 3q³ +6q² + 4q

Hence it is clear that cube of any positive integer is of the form 9m , 9m + 1 or 9m +8.