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Show that n^2-1 is divisible by 8, if n is an odd positive integer. |
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Answer» Explanation:To Prove :n² -1 is divisible by 8, if n is an odd POSITIVE INTEGER.Proof :Let assume that, any odd positive number is in the form of (4R + 1) & (4R + 3) for some integer R. Now,Let, n = 4R + 1=> n² - 1 => (4R + 1)² - 1=> 16R² + 8R + 1 - 1=> 16R² + 8R=> 8R(2R + 1).°. n² - 1 is divisible by 8.Similarly,Let, n = 4R + 3=> n² - 1=> (4R + 3)² - 1=> 16R² + 24R + 9 - 1=> 16R² + 24R + 8=> 8(2R² + 3R + 1).°. n² - 1 is divisible by 8.Hence :n² -1 is divisible by 8, if n is an odd positive integer. |
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