Condition for n² − 1 to Be Divisible by 8

Video Explanation

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Solution

Question: n² − 1 is divisible by 8, if n is:

(a) an integer    (b) a natural number    (c) an odd integer    (d) an even integer

Step 1: Factorise the Expression

n² − 1 = (n − 1)(n + 1)

Step 2: Consider n as an Odd Integer

Let n = 2k + 1, where k is an integer.

Then:

n − 1 = 2k (even)
n + 1 = 2k + 2 = 2(k + 1) (even)

Step 3: Product of Three Consecutive Even Numbers

(n − 1), n, (n + 1) are three consecutive integers.

Among (n − 1)(n + 1), one is divisible by 4 and the other by 2.

∴ (n − 1)(n + 1) is divisible by 8.

Final Answer

✔ n² − 1 is divisible by 8 when n is an odd integer.

✔ Correct option: (c) an odd integer

Conclusion

Thus, whenever n is an odd integer, the expression n² − 1 is always divisible by 8.