Question Use Euclid’s Division Lemma to show that the cube of any positive integer is either of the form 9m, 9m + 1 or 9m + 8. Class X
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you know what is Euclid's Division Lemma
a = bq + r
so let (a) be positive integer
and b = 3;
a = 3q + r
now 0 < or = r < 3
so r = 0, 1 or 2
when r = 0,
(a)³ = (3q + 0)³ = 27q³
= 9m (let m = 3q³)
when r = 1
(a)³ = (3q + 1)³
= 27q³ + 1 + 9q(3q + 1)
= 27q³ + 27q² + 9q + 1
= 9m + 1
(here, let m = 3q³ + 3q² + q )
when r = 2
(a)³ = (3q + 2)³
= 27q³ + 8 + 18q(3q + 2)
= 27q³ + 54q² + 36q + 8
= 9m + 8
(here, let m = 3q³ + 6q² + 4q )
So any positive integer's cube (here a) will be in form 9m, 9m + 1 or 9m + 8
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