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with all of you. So let's get started. In this video i'm going to talk about number properties
and I'll be taking the chapter of even and odd. So let's discuss the concepts
of even and odd in number properties. Let's go with the basic properties of
even numbers. Now even numbers are those which are divisible by 2. Numbers which is 2,
4, 6, 8 that do not leave any remainder when divided by 2 are simply called even numbers so the general
representation or the standard form for all even numbers are is 2a, where a is an integer.
Now I'm just specifying that a is an integer. It can be a positive integer it can be a negative
integer as well and will come to that. Now the difference between two even numbers is always
even. Take any example let's say six and four when you calculate the difference is
equal to two let's take another example 22 minus 10 that's equal to 12. So essentially
the difference between two even numbers is always even in nature. Then zero is an even
the number and a lot of you might have this that zero is neither even nor odd. A lot of people falter
and GMAT tests this a lot. So 0 is an even number, a very important thing that we have to keep in mind.
Next, even numbers can be negative too. so if someone tells you that -2 is not an even number
that's absolutely incorrect. So -2, -4, -6, -12 you know all these numbers all.
Negative numbers which are divisible by 2 are also even numbers. Now that was all about the basic
properties of even numbers so now let's move ahead and look at the basic properties of odd numbers.
Odd numbers are the ones that are not divisible by two. So for example 1, 3, 5, 7, 19, 23 all these
numbers are odd numbers. Now since they're not divisible by 2 the standard form in which we write
an odd number is 2a +1 where a is an integer. Again a is just an integer it can be a positive
integer can be a negative integer. It can be a non-negative integer. for example 0 as well.
For example, i can write 1=2 *0 +1. I can write 3=2*1+1 and similarly, I
can write 7. So all numbers which can be written in the form of 2 a plus 1 where a is an integer
are the odd numbers. Now the difference between two-word numbers are a very important thing and
a lot of students falter in they think that the difference between two odd numbers can be odd
as well. No that's not the case so let's take two examples 5- 3 the difference is equal to 2.
21- 11 the difference is equal to 10. Take any numbers at your own end you would see that
the difference between two odd numbers is always even in nature. A very common faulting point that
students often make when it comes to the GMAT is that they say that
one is the smallest odd number. Now basically this arrives from the fact that people think that odd
numbers can only be positive but odd numbers can be negative two. So minus -1, - 3, -5
-7, all of these numbers are also odd in nature so one is not the smallest odd number.
With that so we had a basic idea about the basic properties of even and odd numbers.
What happens when we apply some mathematical operations on these kind of numbers and this
is something that GMAT tests you on so let's understand the effect of simple operations. For
example addition. Now when you add two even numbers the result is always even in nature. For example
when you add 4+6 the result is equal to 10, when you add an even number with an odd number
a result is an odd number. For example 4 + 3 is equal to 7. When two odd numbers is out are
added a result is an even number so 5 plus 11 is equal to 16. now let's make a combination of that
and let's see how does it affect essentially. even a combination of even and odd numbers would
essentially take all the values or take all the properties that these three steps that I've
written these three pointers that i've written in the start actually follow so let's take an
example of odd plus even plus odd now when i add odd plus even if you look uh at the second point i
have written even plus odd gives me an odd result so odd plus even would be giving me an odd result
with that if i add an odd number look at point number three or plus odd gives me an even number
so then the entire result of odd plus even plus odd gives me an even number for example 3 plus
4 plus 5 when you add 3 and 4 i get 7 and with that if i add 5 i get 12 which is even in nature
what happens when i subtract even and odd numbers now when an even number is subtracted from an even
number you know the resultant is also an even number for example 14 minus 6 is equal to 8.
what happens when i subtract an odd number from an even number 4 minus 1 is equal to 1 i get an
odd result when i subtract two odd numbers i get an even number 15 minus 11 is equal to four and
again if you look at a combination the properties would remain the same so odd minus even minus what
let's first calculate odd minus even it gives me an odd result minus an odd so odd minus odd look at
point number three it would give me an even result take an example for that 13 minus 4 minus 5
13 minus 4 is equal to 9 which is odd minus 5 which is again an odd number gives me a resultant
of 4 which is an even number. Let's take another look at the other operations that that are
there so let's take a look at multiplication for example now when you multiply two even numbers
together you get an even number so 4 times 6 is equal to 24. when you multiply an even number with
an odd number you also get an even number but when you multiply two odd numbers together
the resultant is an odd number well very important thing to note over here is when you multiply an
even number with anything else any other integer number then the resultant is always an even number
and the combination of the same also holds the same properties very interesting what what happens
when i divide even or even odd numbers now when an even number is divided by an even number you
get multiple cases for example if you divide 4 with 6 it would be 1.5 which is not an integer
if you divide 4 by 2 you get an even number which is equal to 2 and if you divide 6 by 2 you get
an odd number which is equal to 3. similarly the same happens when you divide an even number by an
odd number you get multiple cases i've listed down the examples the same also happens for odd numbers
when when i divide an odd number with an odd number i may get a non-integer or i might i might
get end up getting an odd number itself but a very important thing that gmat also tests us all
on is when you divide an odd number with an even number the resultant is always a known integer so
you can take an example and the resultant that you would be getting would always be a non-integer
number at your end with that let's come to the part where gmat tests us the most that is the
effect of exponents what happens when i have even bases now when i have an even base if i raise that
to an even positive integer for example 2 to the power 4 which is an even positive integer then the
resultant is an even number when i raise an even number to the power of an odd positive integer
again the resultant is an even number however there's one small thing that you have to keep
in mind when you take any even number apart from zero and raise it to zero which is again up or
again a non-negative you know even integer it's equal to one which is odd in nature
any even number raised to the power a negative integer can lead to multiple cases for example 2
to the power -1 leads to a non-integer that is half 0 to the power minus 1 gives you 0 which
is which is even in nature so these are the multiple cases that you have to keep in mind
all right now let's start off with the odd basis
now so what happens when i have an odd base for example let me take an example over here
an odd number raised to the power and even positive integer for example 3 raised to the
power 4 gives me 81 so that is odd in nature and odd number raised to the power and odd positive
integer gives me a not result as well for example 3 to the power 3 gives me 27 an odd number raised
to the power 0 for example 3 to the power 0 would also be odd it would give me a resultant of 1.
an odd number raised to the power and a negative integer now that can lead to multiple cases for
example if 3 is raised to -1 then i'll get a non-integer which is equal to 1 by 3. if minus 1
is raised to the power minus 1 the resultant would be an odd number because the resultant would be
equal to minus 1. now based on all of this gmat tests us on three major important things first
is any even number raised to the power of any positive integer is always even in nature any odd
number raised to the power a non-negative integer is an odd number and any integer other than 0
raised to the power 0 is equal to 1 which is odd in nature so please do keep this in mind
all right so now let's understand what are the important things that you need to learn
to master gmat number properties essentially the even odd part the first thing is simplify
the even out questions that you get based on the variables even or nature and not by taking
numbers so do not plug in numbers and check the resultant is an odd number or an even number
you know that can lead you to missing cases and hence can increase the chances of errors
so for example you know the moment you see that in a question you have been asked the even or
property of 3 to the power k and it is mentioned in the question that k is a non-negative integer
it should strike your head that you know 3 to the power k would always be odd if k is a non-negative
integer 2 to the power k if if the question asks what is the even nature of 2 to the power k
then and and in the question it's given that k is a non-negative integer you should evaluate this
and infer that you know 2 to the power k if k is a non-negative integer can be even and can also
be odd because k can also take the value of zero so these are the small inferences that you need to
keep in your head and you need to you need to make to ensure you solve questions in a much accurate
manner the second thing is consider all cases you know most of the students who falter at even odd
questions falter because they fail to consider all cases so this is a very important process skill
and and to know more about this process just you to search on youtube as egmat process skills and
you would get a complete list of videos you know on on on how do you not miss to consider all cases
right so these are the two very important things that you need to keep in mind keep in
your mind to to ensure that you do good in gmat even and odd keep in mind that we will
be posting questions based on even not to test your learning so once you're done with this video
i hope your basics are right at your at their places so make sure you attempt those questions
and to not miss the updates do not miss the video updates that we would be posting
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till the next time, we meet have a very good day ahead and happy learning!