proof by contrapositive or direct proof the following: Please, indicate which pr

Question

proof by contrapositive or direct proof the following: Please, indicate which proof did you use: The square of an odd integer is a number of the form 8t + 1,where t is an integer. Please, can someone write this proof indetail? I need everydetail of it. proof by contrapositive or direct proof the following: Please, indicate which proof did you use: The square of an odd integer is a number of the form 8t + 1,where t is an integer. Please, can someone write this proof indetail? I need everydetail of it.

Explanation / Answer

Note that every odd integer can be written as 4k±1 for somek and some choice of + or -. This dual form representation does thetrick. (4k±1)2 = (4k)2 ±2(4k) +(±1)2 = 16k2 ±8k + 1. Clearly,this is of the form 1 modulo 8 since the first two terms aredivisible by 8.

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