Question 1028881: show that n^2 -1 is divisible by 8, if n is an odd positive integer.
Answer by mathmate(429)   (Show Source):
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Question:
Show that n^2 -1 is divisible by 8, if n is an odd positive integer.
Solution:
If n is odd and positive, we define
n=2k+1 where k is a non-negative integer.
from which we substitute, expand and factor:
n²-1
=(2k+1)²-1
=4k²+4k+1-1
=4k²+4k
=4k(k+1)
Since k is a non-negative integer, we have two possible cases:
1. k is odd, in which case (k+1) is even, and equal to 2q (q=non-negative integer)
=> n²-1=4k(2q)=8kq (where both k and q are non-negative integers)
therefore 8 divides n²-1
2. k is even, then k=2q (q=non-negative integer)
=> n²-1=4(2q)(k+1)=8q(k+1) (where both k and q are non-negative integers)
therefore 8 divides n²-1
Since in both cases, 8 divides n²-1, therefore it is proved that 8 divides n²-1 in for all positive values of n.
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