which is the correct proof of if product of two integers is even then either of the integer is e

Question

Which is the correct proof of if the product of two integers is even, then either of the integers is even? Select one: a. The contrapositive of the statement is if two integers are odd, then the product of those integers must be odd. Suppose a=2k+1 and b=2k+1 are the odd integers, where k is an integer. Then a×b =(2k+1)(2k+1)=4k^2+4k+1=2(2k^2+2k)+1 is odd as 2k^2+2k is an integer. Hence the contrapositive statement is proved. Since the contrapositive and original statement are logically equivalent, the original statement is proved. b. Suppose the product of two integers a and b is even. Then a×b=2k for some integer k. Then a=2 and b=k. Hence one of the integers is even. c. The contrapositive of the statement is if two integers are odd, then the product of those integers must be odd. Suppose a=2k+1 and b=2m+1 are the odd integers, where k and m are integers. Then a×b =(2k+1)(2m+1)=4km+2k+2m+1=2(2km+k+m)+1 is odd as 2km+k+m is an integer. Hence the contrapositive statement is proved. Since the contrapositive and original statement are logically equivalent, the original statement is proved. d. The contrapositive of the statement is if two integers are odd, then the product of those integers must be odd. Suppose a=2k+1 and b=2m+1 are the odd integers, where n and m are integers. Then a×b =(2k+1)(2m+1)=2km+2k+2m+1=2(km+k+m)+1 is odd as 2km+k+m is an integer. Hence the contrapositive statement is proved. Since the contrapositive and original statement are logically equivalent, the original statement is proved.

          Which is the correct proof of if the product of two integers is even, then either of the integers is even?
Select one:
a. 
The contrapositive of the statement is if two integers are odd, then the product of those integers must be odd.
Suppose a=2k+1 and b=2k+1 are the odd integers, where k is an integer. Then a×b
=(2k+1)(2k+1)=4k^2+4k+1=2(2k^2+2k)+1 is odd as
2k^2+2k is an integer. Hence the contrapositive statement is
proved. Since the contrapositive and original statement are logically
equivalent, the original statement is proved.
b. 
Suppose the product of two integers a and b is even. Then a×b=2k for
some integer k. Then a=2 and b=k. Hence one of the integers is
even.
c. 
The contrapositive of the statement is if two integers are odd, then the product of those integers must be odd.
Suppose a=2k+1 and b=2m+1 are the odd integers, where k and m
are integers. Then a×b =(2k+1)(2m+1)=4km+2k+2m+1=2(2km+k+m)+1 is
odd as 2km+k+m is an integer. Hence the contrapositive statement is
proved. Since the contrapositive and original statement are logically
equivalent, the original statement is proved.
d. 
The contrapositive of the statement is if two integers are odd, then the product of those integers must be odd.
Suppose a=2k+1 and b=2m+1 are the odd integers, where n and m
are integers. Then a×b =(2k+1)(2m+1)=2km+2k+2m+1=2(km+k+m)+1 is odd
as 2km+k+m is an integer. Hence the contrapositive statement is proved.
Since the contrapositive and original statement are logically
equivalent, the original statement is proved.

Added by Ana D.

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