Question

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

Answer 1

Answer:

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