Q.1.12 - 1 is divisible by 8, if n is "(A) an integer. (B) a natural number. (C) an odd integer. (D) an even integer,
Answers
Answer:(C) an odd integer
Explanation: Let x = n2 – 1 In the above equation, n can be either even or odd. Let us assume that n= even. So, when n = even i.e., n = 2k, where k is an integer, We get, ⇒ x = (2k)2-1 ⇒ x = 4k2 – 1 At k = -1, x = 4(-1)2 – 1 = 4 – 1 = 3, is not divisible by 8. At k = 0, x = 4(0)2 – 1 = 0 – 1 = -1, is not divisible by 8 Let us assume that n= odd: So, when n = odd i.e., n = 2k + 1, where k is an integer, We get, ⇒ x = 2k + 1 ⇒ x = (2k+1)2 – 1 ⇒ x = 4k2 + 4k + 1 – 1 ⇒ x = 4k2 + 4k ⇒ x = 4k(k+1) At k = -1, x = 4(-1)(-1+1) = 0 which is divisible by 8. At k = 0, x = 4(0)(0+1) = 4 which is divisible by 8 . At k = 1, x = 4(1)(1+) = 8 which is divisible by 8. From the above two observation, we can conclude that, if n is odd, if n odd, n2-1 is divisible by 8. Hence, option (C) is the correct answer.Read more on Sarthaks.com - https://www.sarthaks.com/877775/n-2-1-is-divisible-by-8-if-n-is-a-an-integer-b-a-natural-number-c-an-odd-integer-d-an-even-integer
Answer:
B) A natural number ........