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

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We know that any odd positive integer is of the form 49 +1 or, 4q +3 for some integer q. So, we have the following cases : Case I: When n=4g + 1 In this case, we have n2 - 1 =(4q+ 1)2 - 1 = 16 q2 +8q+1-1 = 1692 +8q=8q (2q+1) = n2-1 is divisible by 8 [: 89 (2q + 1) is divisible by 8] Case II : When n=4q+3 In the case, we have n2 - 1 =(49 +3)2 - 1 = 16q2 + 24q+9-1 = 16q2 +249 +8 = n2-1 = 8(2q2 +39 +1)=8 (2q + 1) (q+1) = n2 - 1 is divisible by 8 : 8(2q + 1) (9+1) is divisible by 87 Hence, n2-1 is divisible by 8.
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