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Show that n2-1 is divisible by 8, if n is an odd positive integer.
Any odd positive integer is of the form 4m + 1 or 4m + 3 for some integer m. When n = 4m + 1, n2 – 1 = (4m + 1)2 – 1 = 16m2 +8m+1 – 1= 16m2+8m = 8m(2m+1) n2 – 1 is divisible by 8. When n = 4m + 3 n2 – 1 = (4m+3) – 1 = 16m2 + 24m + 9 – 1 = 16m2 + 24m + 8 = 8(2m2 + 3m + 1) n2 – 1 is divisible by 8. Hence, n2 – 1 is divisible by 8 if n is an odd positive integer.
n2 – 1 = (4m + 1)2 – 1 = 16m2 +8m+1 – 1= 16m2+8m = 8m(2m+1) n2 – 1 is divisible by 8. When n = 4m + 3 n2 – 1 = (4m+3) – 1 = 16m2 + 24m + 9 – 1 = 16m2 + 24m + 8 = 8(2m2 + 3m + 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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