Examples of Fundamental theorem of Arithmetic Video Lecture - Class 10

hello friends welcome welcome to this
presentation from rising pearl I'm a
Friendly host Roy friends today we are
talking about series 1 where we are
discussing real numbers
this is webisode number 9 and today's
topic is solving number property related
questions using fundamental theorem of
arithmetic the friends in the last
episode we have taken a detail look at
what do we mean by fundamental theorem
of arithmetic so in a nutshell what the
fundamental theorem of arithmetic states
is this every composite number can be
expressed as product of primes in one
and only one unique way if we do not
consider the order in which the prime
numbers appear so that is if you take a
composite number X let's say we take a
composite number X then X can be written
as product of primes so P 1 P 2 P 3 we
multiply P 1 times P 2 times P 3 etc so
where each of them they are actually
prime numbers so we can express every
composite number so for example we know
like 6 is a composite number so we can
write 6 as 2 times 3 where both 2 and 3
are prime so we can write 6 is this
similarly if you take a look at the
number 9 we can write 9 as 3 times 3
that is we can express minus product of
prime again right so something like this
we will probably write this as 3 to the
power 2 so similarly let us say if we
have a prime number a composite number
24 we can write this as 2 times 18 sorry
2 times 12 right and then for 12 we can
write this as 2 times so 12 we can write
this as 4 times 3 and then 4 we can
write it as 2 times 2 so there will be 3
2 so 2 times 2 times 2 so we will have 2
to the power 3 multiplied by 3
so friends this is something we have
seen net length in the last webisode
which is that every single composite
number we can express them as product of
prime numbers right so let's find out
how do we apply this knowledge to solve
some of the problems and what kind of
problems are we talking about so the
question is say a number is represented
by 4 to the power n where N is a natural
number can we say that the number for n
ever end with 0 for any value of n so
friends read the question one more time
so what the question is saying is we
don't have a number but we have a format
for the number which is 4 to the power N
and n can have any value natural number
we know is a collection of all the
numbers like 1 2 3 4 like this to plus
infinity right so n can have any value
so the question is saying is there a
value of n for which this number will
end with 0 that is so for example if n
is 1 if n is 1 our number is 4 right if
n is 2 our number is 4 times 4 which is
16 right so similarly for N equals 3 we
will have 16 for 64 so question is
saying is there a value of n remember n
is a collection of natural numbers it
can have any value so question is saying
is there a value of n for which this
entire number can end with 0 that means
there is if there is a value of n I can
write for n as something and it ends
with a 0 so is it possible that is the
question so friends the first thing you
can think about is so if the number ends
with 0 right so something like maybe 10
or 20 or 30 or something right if you
notice if the number ends with 0 what
jumps out to us
well they are all divisible by 10
or they are all divisible by five right
each of these numbers I can divide by
five and why am I saying five and not
some other number and you will get to
know that in a moment each of the
numbers if the end with zero they are
all divisible by 10 but they are also
divisible by five why 5 because 5 is a
prime number that is why we are
interested in 5 so if there is a value
of n for which 4 n ends with 0 it means
that that number must be divisible by 5
in other words this number will have 5
as one of its factors right so for
example 10 because 10 can be divided by
5 we can write 10 as 2 times 5 right so
if a number is divisible by 5 it means a
number itself has 5 as one of its
factors so what we are saying is for 4 n
must have a factor 5 right but let's
take a look at this for n is nothing but
we can write for n as we can write 4 as
2 to the 2 times 2 to the power n or we
can write this as 2 square power n or
this means 2 to the power 2 N and what I
did here here and here is just the basic
laws of exponents right so what we see
is this is very interesting
so what we find is for n equal to 1 this
value will be 2 n equal to 3 this will
be 6 so it means that the only prime
factor that we can see for a number for
n is 2 because if you take n equal to 1
my number my number like if it is 4
equal to 1 right my number is 2 if n
equal to say 3 my number 3 is 6 so no
matter what the value of n is I have
only 2 as my factor I do not see 5 and
so we say that for n can never end with
0 because 5 can never be a prime factor
4 for n because the fundamental theorem
of arithmetic states that for n can be
represented in only one and one unique
way in terms of product of its rights so
if I have found out that foreign can be
written as 2 to the power 2 n and this
means only 2 is the only prime factor we
can never have 5 as a factor 4 for N and
so it can never end with 0 so Frank's
how do we write this question if it
comes up in a test so we start with a
question and then what we say is let's
assume that for some value of n 4 and
ends with 0 then for n will be divisible
by 5 so if we find out the prime factors
of 4 & 5 will be one of the factors
right because for n is divisible by 5
clearly means that 5 is a factor right
but then for N equals 2 times 2 to the
power n or essentially it means 2 to the
power 2 n which means from fundamental
theorem of arithmetic we know that 2 is
the only prime factor of foreign for any
value of N and hence 5 cannot be a
factor so foreign cannot end with 0 for
any value of n so things will take a
look at one more similar question and I
am NOT going to get into details of how
you write it but I'm just going to give
you a general idea so the question in
this case is is 6 to the power n where N
is a natural number can this number ever
end with 0 so again similar difference
you can say that 6 n can be written as
basically 2 times 3 right 2 times 3 to
the power n which means 2 to the power n
multiplied by 3 to the power n so again
if 6n can have if 6n can be written as
any number that ends with 0 that means
this number should be divisible by 5
right
that means this number should be we can
we should be able to express this number
as something multiplied by 5 because 5
should be a factor right if we get a 0
at the end that means that this number
can be divided by 5 or it means that
this number can be expressed as
something times 5 where 5 should be one
of the factors but we just saw that 6n
has only 2 to the power N and 3 to the
power N and fundamental theorem of
arithmetic says that this is the only
way we can express 6 to the power n so
if n equals to 5 for example the factors
will be 2 to the power 5 and 3 to the
power 5 there is no 5 here if n equals
if n is 10 for example 6 to the power 10
whatever that number is that number will
be written as 2 to the power 10 times 3
to the power 10 so 5 will never be a
factor and so we say that 6 to the power
n can never end with 0 so friends in the
next webisode we will continue to look
at some more applications of fundamental
theorem of arithmetic thanks for
watching