show that n square minus 1 is divisible by 4 if n is an odd positive integers
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Any odd positive integer is the form of 2m + 1 or 2m + 3 for some integer m.
When n = 2m + 1,
n^2-1=(2m+1)^2-1=4m^2+4m+1-1=4m^2+4m=4m(m+1)
⇒ n^2 – 1 is divisible by 4.
when n =2m+3
n^2-1= (2m+3)^2-1=4m^2+12m+9-1=4m^2+12m+8=4(m^2+3m+2)
Hence, n^2- 1 is divisible by 4 if n is an odd positive integer.
When n = 2m + 1,
n^2-1=(2m+1)^2-1=4m^2+4m+1-1=4m^2+4m=4m(m+1)
⇒ n^2 – 1 is divisible by 4.
when n =2m+3
n^2-1= (2m+3)^2-1=4m^2+12m+9-1=4m^2+12m+8=4(m^2+3m+2)
Hence, n^2- 1 is divisible by 4 if n is an odd positive integer.
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