use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m +8 Related: Fundamental Theorem of Arithmetic (Hindi)?

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 Related:  Fundamental Theorem of Arithmetic (Hindi)?

Answer

Answer

According to Euclid's division Lemma ; a = bq + r So b = 3 Let r = 0, 1, 2 , 3 ... so on . In first case a = bq + r a = 3q + 0 a = ( 3q )^3 a = 27 q^3 a= ( 3) (3) (3q^3) where 3q^3 = m So , a = 9m In second case Now put the value of r = 1 a = bq + r a = 3q + 1 a = ( 3q + 1)^3 Using formula ( a + b )^3 = a^3 + b^3 + 3ab ( a + b) a = 27q^3 + 1 + 9q ( 3q +1 ) a = 27q^3 + 1 + 27 q^2 + 9q a= 27q^3 + 27q^2 + 9q + 1 a = 9 ( 3q^3 + 3q^2+ 1 ) + 1 where ( 3q^3 + 3q^2 + 1) = m a = 9m + 1 In third case Now put the value of r = 2a = bq + r a = 3q + 2 a = ( 3q + 2 )^3 a = 27q^3 + 8 + 18q ( 3q + 2 ) a = 27q^3 + 8 + 54q^2 + 36q a = 27q^3 + 54q^2 + 36q + 8 a = 9 ( 3q^3 + 6q^2 + 4q ) + 8 where ( 3q^3 + 6q^2 + 4q ) = m a = 9m + 8 Hence proved !!

Answer

Euclid's Division Lemma:
Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < />

Proving that the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8:

Step 1: Breaking down the problem
To prove this statement, we need to show that for any positive integer n, the cube of n can be expressed in the form 9m, 9m + 1, or 9m + 8. Let's consider each case separately.

Case 1: n is divisible by 3
When n is divisible by 3, we can write n as n = 3k, where k is an integer. The cube of n can be written as n³ = (3k)³ = 27k³. Here, we can see that the cube of n is of the form 9m, where m = 3k³.

Case 2: n leaves a remainder of 1 when divided by 3
When n leaves a remainder of 1 when divided by 3, we can write n as n = 3k + 1, where k is an integer. The cube of n can be written as n³ = (3k + 1)³ = 27k³ + 27k² + 9k + 1. Here, we can see that the cube of n is of the form 9m + 1, where m = 3k² + 3k + 1.

Case 3: n leaves a remainder of 2 when divided by 3
When n leaves a remainder of 2 when divided by 3, we can write n as n = 3k + 2, where k is an integer. The cube of n can be written as n³ = (3k + 2)³ = 27k³ + 54k² + 36k + 8. Here, we can see that the cube of n is of the form 9m + 8, where m = 3k² + 6k + 3.

Step 2: Proving the statement
From the above cases, we have shown that for any positive integer n, the cube of n can be expressed in the form 9m, 9m + 1, or 9m + 8. Therefore, the statement is proven.

Conclusion:
Using Euclid's Division Lemma, we have shown that the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8. This result is useful in number theory and provides insight into the properties of cubes of positive integers.

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