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?