Hello I am having some trouble trying to know which way I should go as for this proof: ( we are suppose to use either direct, contradiction or contraposition)
Prove or disprove: If the product of two integers is even, at least one of them must be even.
Im thinking the direct method of proof would work so heres my start:
let a, b be PBAC even integers
let a = 2m
let b = 2n
the product of 2m and 2n is 4mn
4mn= 2(2mn)
because their both integers, their product must be an integer. Sense we multipling 2mn by 2 it is also even.
now lets have one of them be odd:
let a be PBAC even integer
let b be PBAC odd integer
let a = 2m
let b = 2n + 1
the product: 2m(2n +1) = 4mn + 2n = 2(2mn +1)
m and n are integers so the product is an integer.
adding 1 is also an integer but it is in the form of a multiple of 2, so it is even.
so for a product to be even, and integer must be even
QED?
Is my proof correct? and how would i do this by contradiction/contraposition
My attempt at contradiction:
Assume the product of two integers a and b are both even. Suppose not that integers a and b are odd
a = (2m + 1)
b = (2n + 1)
let m and n be PBAC integers
ab = (2m + 1)(2n + 1)
ab = 4mn + 2m + 2n +1
ab = 2(2mn + m + n) + 1
Since (2mn + m + n) will be an integer by closure we can conclude that the product ab is an odd integer, which contradicts the assumption.
This means at least one of the integers a or b must be even.
QED?
Best Answer
Your proof isn't quite correct; you've concluded that if one of the integers is even, so is their product. You want to prove that it the product is even, then one of the integers was even.
The proof by contrapositive is the simplest, I think. The contrapositive simply says that
That is, the product of two odd integers is odd. This is easily verified from considering the equality
$$(2k + 1)(2m + 1) = 4km + 2k + 2m + 1 = 2(2km + k + m) + 1$$