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7.Showthat2- 1 is divisible by 8, if n is an odd positive integer.18. If the zeroes of the nolvnomial 3 321 are a-h a'nt h find a and

Answer»

Ans :- Given⇒ n is an odd positive integer.Proof⇒We know that an Odd Positive Integer nis always written in the form of (4k + 1) or (4q + 3) or (4q + 5) and so on.

If n = 4k + 1Then, n² - 1 = (4k+ 1)² - 1= 16 k² + 1 + 8k - 1[∵(a + b)² = a² + b²+ 2ab]= 16k² + 8k= 8k(2k + 1)

Hence, it is divisible by 8.

If n = 4k + 3Then, n² - 1 = (4k + 3)² - 1= 16k² + 9 + 24k - 1= 16k² + 24k + 8= 8(2k² + 3k + 1)

Hence, it is also divisible by 8.

If n = 4k + 5Then, n² - 1 = (4k + 5) - 1= 16k² + 25 + 40k - 1= 16k² + 40k - 24= 8(2k² + 5k - 3)

Hence, it is also divisible by 8.

Now, For any value of n, n² - 1 is always be divisible by 8.


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