Waching daily Feb 2 2018

It's New Year's, and you brought the younger one here too.

I can't take this anymore.

I am her mom.

I brought her here for New Year's. What's the problem?

I'm not going to raise someone else's kid.

I'm Wenya's father. I should take care of her.

Where would I send her to?

Her own mother doesn't want her.

Why do I have to let her in?

It's not your home, no one wants you!

Jingjing, why is it that we have two homes when other kids seem to only have one?

Hm … Mama said that we'll understand when we're older.

Dad, what's wrong?

Your mother's in the hospital.

Wenya, one day if I'm not here anymore,

you need to take care of yourself.

So no matter how hard things get, I know we can all get through it.

Why couldn't, why couldn't I have had a family that was whole?

Auntie, why is life always so full of suffering?

Almighty God's words very clearly explain the root of people's pain in life.

Let's take a look at His words together. Then you'll see.

You need to know that God is with us now.

The Church is our true family.

We're no longer alone.

For more infomation >> God Is Here | Best Christian Movie Trailer "Where Is My Home" | A True Story - Duration: 2:50.

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"DA CIRCLE"...JACKEE' HARRIS ON SEXUAL ASSAULT,HER ### IS CRAZY!!! LMAO - Duration: 6:49.

For more infomation >> "DA CIRCLE"...JACKEE' HARRIS ON SEXUAL ASSAULT,HER ### IS CRAZY!!! LMAO - Duration: 6:49.

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Bitconnect is Not a Scam!!!!! - Duration: 14:46.

For more infomation >> Bitconnect is Not a Scam!!!!! - Duration: 14:46.

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Finding the least perfect square that is divisible by a set of numbers - Duration: 6:12.

Hello and welcome to the "Magic of Mathematics" channel.

My name is Manohar Moorthy and you are watching one amongst a series of videos exploring

the world of Mathematics.

In this video, I will show you how to find the least perfect square that is exactly divisible

by a set of numbers.

For this, I will take an example; say, we have been asked to find the least perfect square

that is exactly divisible by the numbers 6, 9, 12 and 15.

I will begin by observing that the least perfect square that is exactly divisible by a set of numbers

can not be lesser than the least number that is exactly divisible by the given numbers.

And we know that the least number that is exactly divisible by a set of numbers is the

lowest common multiple or LCM of the given numbers.

Hence, I will find the LCM of the given numbers first and see if it is a perfect square;

if it happens to be a perfect square we have the least perfect square that is exactly divisible

by the given numbers and if not, I will use the LCM as a base or starting point for finding

the number that I am looking for.

For finding the LCM, I will prime factorise each of the given numbers.

6 will prime factorise as 2 times 3, nine prime factorises as 3 times 3, 12 as 2 times 2 times 3

and 15 as 3 times 5.

For each factor in our prime factorisations, that is 2, 3 and 5, I will identify its occurrences

in the prime factorisation where it occurs the maximum number of times.

2 occurs maximum number of times here, 3 here and 5 here.

Now, I can find the LCM by computing the product of the identified occurrences of the factors

or in other words, the LCM of the given numbers is 2 times 2 times 3 times 3 times 5.

It is clear that if I compute this product of prime factors, I will get the LCM of the

given numbers.

I will now examine this product to see whether the LCM is a perfect square.

For this, I will use the fact that if a number is a perfect square, its prime factorisation

can be grouped into pairs of identical factors.

Examining the prime factorisation of the LCM, we can see that we have a pair of 2's and

a pair of 3's.

However, there is only one 5 and hence we can't form a pair of 5's.

Hence, this product which corresponds to the LCM of the given numbers is not a perfect square.

This means that the least perfect square that we are looking for has to be greater

than the LCM.

But since it also has to be exactly divisible by the given numbers, or in other words, since

it is a common multiple of the given numbers, it has to be a multiple of the LCM.

Hence, I will examine successive multiples of the LCM to see whether they are perfect squares

and once I get a perfect square, I will stop since I would have obtained the

least perfect square that is exactly divisible by the given numbers.

I will begin with the least non-trivial multiple of the LCM, namely twice the LCM.

Its prime factorisation can be obtained by multiplying the prime factorisation of the LCM by 2.

As we can see, now in addition to the 5, there is a 2 which is not paired up.

So, we still don't have a perfect square.

The next multiple of the LCM, namely 3 times the LCM has an unpaired 3 in addition to the

unpaired 5.

So, this will also not work.

The next multiple of the LCM, namely 4 times the LCM has 2 additional 2's in its prime factorisation

and these two form a pair but we still have an unpaired 5.

So, this will also not work.

The next multiple of the LCM, namely 5 times the LCM finally supplies the 5 required for

the 5's to pair up.

We now have a product which will give us the least perfect square that is exactly divisible

by the given numbers.

If we compute this product, we get 900 as our answer.

Or in other words, 900 is the least perfect square which is exactly divisible by the given numbers

A closer look at what I did gives the method which can be used for solving such problems.

Basically, we obtain the prime factorisation of the LCM of the given numbers and form pairs

of identical factors as shown to see whether the LCM is a perfect square.

In case nothing gets left out in the pairing process, the LCM is a perfect square and we

have the least perfect square which is exactly divisible by the given numbers.

In case something gets left out as in this case where the factor 5 got left out, we supply

what is required for perfect pairing - an additional 5 in this case takes care of the pairing

so that we obtain the least perfect square that is exactly divisible by the given numbers.

In this video, I showed you how to find the least perfect square which is exactly divisible

by a given set of numbers.

If you found this video useful, I would appreciate a like.

Also, you can subscribe to my channel to be notified as and when I publish more such videos.