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Show that n2−1 is divisible by 8, if n is an odd positive integer.

Show that n2−1 is divisible by 8, if n is an odd positive integer.

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 60787 Points
2 years ago
So, we know that odd positive integers are 3,5,7,.. Read the question once again. So we have to square an odd positive integer and subtract 1 from it, right? Let's look at example, We’ll take 3 So, 32 is 9 and 9-1=8. Similarly, We’ll take 5 So, 52 is 25 and 25-1=24 If you look at these two examples, 8,24 are divisible by 8. Hence, we can easily conclude that n2−1 is divisible by 8 for any odd positive odd integer. Complete step by step answer: Any odd positive number is in the form of (4p + 1) or (4p + 3) for some integers p. Let, n = 4p + 1 n2−1 = (4p+1)2 - 1 On simplifying RHS we’ll get, 16p2+1+8p−1 ⇒ 8p(2p + 1) ⇒n2−1 is divisible by 8 Now, let n = (4p + 3) n2−1=(4p+3)2−1 ⇒16p2+9+24p−1 ⇒16p2+8+24p ⇒8(2p2+3p+1) ⇒n2−1 is divisible by 8 Hence, n2−1 is divisible by 8 if n is an odd positive integer. NOTE: - In this type of question, always write the general points of odd positive integers and then think practically and solve them. First, understand the problem correctly, then the solution becomes easy for you.

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