Question

plz help me with the questions.
it's urgent.
Source: RS Agarwal 2019-20
Class 10​

Answer 1

Question

For any positive integer n, prove that n³ - n is divisible by 6

Solution

To show that (n³ - n) is divisible by 6, we will have to show that (n³ - n) is divisible both by 2 and 3.

n³ - n can be written as

n(n² - 1)

⇒ n(n + 1)(n - 1)

Now, n can either be even or odd

If n is even, then n is divisible by 2

Let n = 2m

⇒ n(n + 1)(n - 1) = 2m(n + 1)(n - 1)

2m(n + 1)(n - 1)is divisible by 2

⇒ n(n + 1)(n - 1) is divisble by 2

If n is odd, then let n = 2m + 1

⇒ n - 1 = 2m + 1 - 1 = 2m

n + 1 = 2m + 1 + 1 = 2m + 2 = 2(m + 1)

Thus, both (n - 1) and (n + 1) are divisible by 2

Hence,

⇒ n(n + 1)(n - 1) is divisible by 2.

So, for any positive integer n, n(n + 1)(n - 1) is divisible by 2

Now, n can be of the form of 3m, 3m + 1 or 3m + 2 (Euclid's division lemma)

When n = 3m, then

n(n + 1)(n - 1) is divisible by 3

When n = 3m + 1,

n - 1 = 3m + 1 - 1 = 3m

⇒ (n - 1) is divisible by 3

⇒ n(n - 1)(n + 1) is divisible by 3

When n = 3m + 2

⇒ n + 1 = 3m + 2 + 1 = 3m + 3 = 3(m + 1)

This means that n + 1 is divisible by 3

⇒ n(n - 1)(n + 1) is divisible by 3

hence, for any positive integer n, n(n + 1)(n - 1) is divisible by 3

Since n(n + 1)(n - 1) is divisible by both 2 and 3, it is divisible by 6 also

⇒ n(n + 1)(n - 1) is divisible by 6

⇒ n(n² - 1) is divisible by 6

⇒ n³ - n is divisible by 6.

Proved.  

Question

Prove that if x and y are both odd positive integers then x² + y² is even but not divisible by 4

Solution

Since both x and y are odd positive integers, let

x = 2n + 1

y = 2m + 1 for some integers n and m

So, x² + y² = (2n + 1)² + (2m + 1)²

Now, using (a + b)² = a² + b² + 2ab

⇒ 4n² + 4n + 1 + 4m² + 4m + 1

⇒ 4n² + 4n + 4m² + 4m + 2

⇒ 2(2n² + 2n + 2m² + 2m + 1)

⇒ 2p (for some integer p = 2n² + 2n + 2m² + 2m + 1)

Since 2p is divisible by 2, it's even

⇒ 2(2n² + 2n + 2m² + 2m + 1)  is even

⇒ (2n + 1)² + (2m + 1)² is even

⇒ x² + y² is even.

But since we couldn't factor 4 from 4n² + 4n + 4m² + 4m + 2 this expression, it is not divisible by 4. Hence, x² + y² is not divisible by 4.

Proved.