Question

show by direct proof,If n is odd, then (n
^2− 1) is a multiple of 8.​

Answer 1

Answer:

>>  n = 2m + 1

>>  n + 1 = 2m + 2

>>  n  −  1  =  2m

>> n^2 −  1  =  (n+1)  * (n−1)  =   (2m + 2)  *  (2m)  =    4m * (m+1)

>> at least one of m,m+1 is even

    thus,

>>  m * ( m + 1 )  =  2k  

      then  

>>   n^2  − 1   =  4 (2k)  =  8k  ⟹  8 ∣ n^2  −  1

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