Use the quotient-remainder theorem with divisor equal to 2 to prove the following statement. a. The square of any integer can be written in one of the two forms 4k or 4k+ 1 for some integer k.

Please read the enswer below

Step-by-step explanation:

The quotient-remainder theorem establishes that given an integer n, there are unique integers d and r, with 0≤r<d, and such that:

q: the quotient

d: the remainder

d: divisor = 2

By the quotient-remainder theorem with divisor 2, you have:

n = 2q

n = 2q + 1

Then, for both cases you have:

  (the square of an integer with divisor 2 is 4k)

with

but 2q + 1 = n

where you have taken   (the product 2qn is another integer)

goog may have it for you