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

$\large\underline{\sf{Solution-}}$

Let a be any positive integer.

Then,

• By Euclid Division Lemma, corresponding to positive integers a and 3, there exist unique integers q and r such that a = 3q + r, where r = 0, 1, 2.

Now, 3 cases arises,

Case :- 1

$\rm :\longmapsto\:When \: a \: = \: 3q$

So,

$\rm :\longmapsto\: {a}^{3} = {(3q)}^{3}$

$\rm :\longmapsto\: {a}^{3} = {27q}^{3}$

$\rm :\longmapsto\: {a}^{3} = {9 \times 3q}^{3}$

$\bf :\longmapsto\: {a}^{3} = 9m \: \: \: \: where \: m \: = {3q}^{3}$

Case :- 2

$\rm :\longmapsto\:When \: a \: = \: 3q + 1$

So,

$\rm :\longmapsto\: {a}^{3} = {(3q + 1)}^{3}$

$\rm :\longmapsto\: {a}^{3} = {27q}^{3} + 1 + 3 \times 3q \times1 (3q + 1)$

$\: \: \: \: \: \: \: \: \: \boxed{ \purple{\sf \because \: {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}}$

$\rm :\longmapsto\: {a}^{3} = 27 {q}^{3} + 1 + 9q(3q + 1)$

$\rm :\longmapsto\: {a}^{3} = 27 {q}^{3} + 1 + {27q}^{2}+ 9q$

$\rm :\longmapsto\: {a}^{3} = 27 {q}^{3}+ {27q}^{2}+ 9q + 1$

$\rm :\longmapsto\: {a}^{3} = 9(3{q}^{3}+ {3q}^{2}+ q) + 1$

$\bf:\longmapsto\: {a}^{3} = 9m + 1 \: \: \: \: where \: m = 3{q}^{3}+ {3q}^{2}+ q$

Case :- 3

$\rm :\longmapsto\:When \: a \: = \: 3q + 2$

So,

$\rm :\longmapsto\: {a}^{3} = {(3q + 2)}^{3}$

$\rm :\longmapsto\: {a}^{3} = {27q}^{3} + 8 + 3 \times 3q \times2 (3q + 2)$

$\: \: \: \: \: \: \: \: \: \boxed{ \purple{\sf \because \: {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}}$

$\rm :\longmapsto\: {a}^{3} = 27 {q}^{3} + 8 + 18q(3q + 2)$

$\rm :\longmapsto\: {a}^{3} = 27 {q}^{3} + 8 + {54q}^{2}+ 18q$

$\rm :\longmapsto\: {a}^{3} = 27 {q}^{3} + {54q}^{2}+ 18q + 8$

$\rm :\longmapsto\: {a}^{3} = 9(3{q}^{3} + {6q}^{2}+ 2q) + 8$

$\bf :\longmapsto\: {a}^{3} = 9m+8 \: \: \: where \: m=3{q}^{3} + {6q}^{2}+ 2q$

Hence,

• Cube of any positive integer is of the form 9m or 9m + 1 or 9m + 8.

Additional Information :-

Fundamental Theorem of Arithmetic :-

• This theorem states that Every composite number can be factorized as the product of their primes and this factorization is unique irrespective of their places of prime.

Euclid Division Lemma :-

• Let a and b are two positive integers such that a > b, then there exist unique integers q and r such that a = bq + r where, r = 0, 1, 2, _____, r - 1.