Even and Odd Integers Multiplication Combinations

Question

Hello,

The possible combinations for multiplying two integers are:

even x odd
even x even
odd x odd

These are the 3 possible combination; how did we come up with that mathematically? N! formula doesn't work, neither does the combination or permutation formula. 
Please help.

Thanks.

OptimusPrepJanielle · Answer

Consider it in the following way:

Even number will be of the form: 2k
Odd number will be of the form: 2k + 1

even x odd = 2k * (2k + 1)
Whatever be the value of k, this will always have 2 as a factor. Hence even

even x even = 2k * 2k = 4 k^2 
Whatever be the value of k, this will always have 2 as a factor. Hence even

odd x odd = (2k + 1)*(2k + 1) = 4 k^2 + 4k + 1 = 4k*(k + 1) + 1
The first part will always be even and when 1 (an odd number) is added to this, the number becomes odd.

Does this help?

zigzag91 · Answer

I truly appreciate the effort you've put in your input, however, you've got my question wrong. What I am talking about is combinations, as in (ab, bb, aa, and ba not counting here). I know it is very simple to solve, but I can't see other way of looking at it from this perception: (2X2)-1, -1 because even x odd is the same as odd x even. What scares me is that this simplified vision wont work at harder problems unless I understand the concept here.

Thanks

Even and Odd Integers Multiplication Combinations

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