Math, asked by shukla19harshita, 8 hours ago

please answer this two questions any one please don't spam ​

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Answered by kaurv7444
5

7) Let us consider

a = n3 – n

a = n (n2 – 1)

a = n (n + 1)(n – 1)

Assumtions:

1. Out of three (n – 1), n, (n + 1) one must be even, so a is divisible by 2.

2. (n – 1) , n, (n + 1) are consecutive integers thus as proved a must be divisible by 3.

From (1) and (2) a must be divisible by 2 × 3 = 6

Thus, n³ – n is divisible by 6 for any positive integer n.

8)Let us consider two odd positive numbers be x and y where

x = 2p + 1 and y = 2q + 1

From question,

x2 + y2 = (2p + 1)2 +(2q + 1)2

= 4p2 + 4p + 1 + 4q2 + 4q + 1

= 4p2 + 4q2 + 4p + 4q + 2

= 4 (p2 + q2 + p + q) + 2

Form above result, we can conclude that x and y are odd positive integer, then x2 + y2 is even but not divisible by four.

Hope that helps you army .

Answered by priyankaanneherenj
1

Answer:

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