# Numbers Properties || Operations of Numbers || Even + Even || Even + Odd || Odd + Odd || Odd+Even…

in this lesson we’ll examine a
systematic technique for drawing conclusions about expressions involving
even and odd numbers to demonstrate this technique we’ll examine a question we
saw earlier now in this question we must determine whether X is even and the two
statements involve expressions that evaluate to be odd numbers let’s begin
with statement 1 it tells us that XY plus y is odd now this expression
involves two variables x and y and each variable can be either odd or even so
there are several different cases to consider X and y can both be odd or X
can be even and y can be odd and so on now each of these cases will cause the
expression XY plus y to evaluate to be either odd or even to keep track of the
different cases we’ll use a table and for each case we’ll determine whether
the expression evaluates to be odd or even so what are the possible cases here
well x and y can both be even X can be even and y can be odd X can be odd and Y
can be even or both variables can be odd so there are four different cases to
consider now let’s see what effect each case has on our expression XY plus y so
for the first case we’ll replace the XS and the Y’s with E to show that they are
both even to evaluate this we’ll apply our rules regarding even and odd numbers
so in the first case we have an even number times an even number which is
even and to this we add an even number which will result in an even number so
for the first case where X and y are both even the expression XY plus y
evaluates to be even for the next case will plug in YZ and those for the even
and odd numbers when we apply our rules we see that the expression evaluates to
be an odd number we’ll follow these same steps for the other two cases as well so
we now know the outcomes in each of the four cases statement 1 tells us that the
expression evaluates to be an odd number so when we examine our table we see that
there is only one case where the expression evaluates to be an odd number
in this particular case X is even and Y is odd so since we can now be certain
that X is even we can answer the target question which means statement one is
sufficient now some students may find the cumbersome to plug YZ and O’s into
the expression and then apply these rules for each case so another strategy
is to plug in actual numbers into our expression for example if x and y are
both even we can replace both variables with twos when we do this our expression
evaluates to be 6 which is even now it doesn’t matter which odd numbers and
which even numbers you plug in however it’s a good idea to use small numbers to
make our calculations easier so for odd numbers plugging in a one will make
calculations easy and for even numbers 2 is a good number to use now using a zero
for even numbers can make calculations even easier however the only drawback is
that a zero might get confused with an O which represents odd numbers the
important point here is to use small values when plugging in numbers ok for
the next case we’ll plug in a 2 for the even number and a 1 for the odd number
when we do this our expression evaluates to be 3 which is odd and we can continue
this method with the other two cases and when we do so we see that our results in
this table are the same as they were in the first table only one case results in
the expression evaluating to be an odd number so once again we can be certain
that X is even which means statement one is sufficient okay now want the
statement to we’ll use the same strategy to examine the four possible cases so
for the first case we’ll plug in YZ to represent even numbers and then apply
our rules to see that the expression evaluates to be an even number following
the same steps we can test the other three cases as well alternatively we can
plug in even and odd into our expression and then evaluate
them to get identical results we have now tested all possible cases
now statement 2 tells us that the expression evaluates to be an odd number
when we examine either of our tables we see that there are two possible cases
where the expression evaluates to be an odd number in one case X is even and in
the other case X is odd so since X can be either even or odd statement two is
not sufficient which means the correct answer here is a so we now have a
systematic technique for drawing conclusions about expressions involving
even and odd numbers so when you encounter a question involving even and
odd numbers consider creating a table to test the various cases if you use a
table you can let either those represent even and odd numbers and then apply the
following rules or you can plug even and odd numbers into the expression and then
evaluate finally once you have tested each case you can draw conclusions based
on the various outcomes