Math tips!!!
by Shreyas 9A

What are H.C.F. and L.C.M.?

HCF: Highest Common Factor

LCM: Least Common Multiple

As you can see from the names, H.C.F. is the greatest factor that any number of numbers have in common. L.C.M. is the least multiple that any number of numbers has in common.

(Just in case you’ve forgotten, factors are numbers that can be multiplied with other factors to get a certain number. For example, since 6 times 5 equals 30, 6 and 5 are factors of 30.)
(And multiples are numbers that are equal to a certain number times some other number. For example, since 8 equals 2 times 4, 8 is a multiple of 2)

So this means on the multiplication tables, all the numbers you have in one table—like the 6 table is: 6, 12, 18, 24….—are all multiples, meaning 12, 18, 24 are all multiples of 6, and 6 is a factor of these numbers.

Anyways, back to H.C.F. and L.C.M.…

HCF is a number such that the numbers that have that HCF are its multiples.
And LCM is a number such that the numbers that have that LCM are its factors.

Does that sound confusing?

If it does, let me show you an example…

Find the H.C.F of 30 and 45.
First, we find the factors of these numbers.

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
(All these numbers can be multiplied with another one of these to get 30)

Factors of 45: 1, 3, 5, 9, 15, 45
Now let’s look at what factors are common.
Clearly 1, 3, 5, and 15 are common factors for 30 and 45.
But the largest one is 45, so the H.C.F. of 30 and 45 is 15.

Now let’s try L.C.M…

Find the L.C.M. of 6 and 8.
First, list some multiples for each of the numbers.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60…

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80…

As you can see from the first 10 multiples of 6 and 8, 24 and 48 are common multiples.

But since 24 is less, the L.C.M. of 6 and 8 is 24.

Is that ALL?

Well, the method we used to find the LCM of 6 and 8 worked well, but what happens if we have bigger numbers, where listing out the multiples might be harder?

Well, here’s a cool little equation…

1st number * 2nd number = HCF * LCM

Take any two numbers and you’ll find it’s true.

Let’s try it out for 6 and 8.
We already know that the L.C.M. of 6 and 8 is 24, so let’s quickly check the H.C.F. .

Factors of 6: 1, 2, 3, 6
Factors of 8: 1, 2, 4, 8

The H.C.F. is clearly 2.
Now, 6 is the 1st number, 8 is the 2nd number, 24 is the L.C.M., and 2 is the H.C.F. .

Which means that…

6 * 8 = 2 * 24
48 = 48
It’s true!!!
This formula can be helpful in some cases.

There is also a simple method called common division method that is useful if you know the divisibility rules.

Suppose I want to find the H.C.F. of 40, and 56.

|_40,_56__

Just find a common factor (preferably small) that can divide both numbers.
For 40 and 56, you can take 2. Write it like this.

2|_40,_56__
ÂÂÂÂ ÂÂÂÂ ÂÂÂÂ ÂÂÂÂ 20,28

You can divide them by 2 again.

2|_40,_56__
2|_20,_28__
ÂÂÂÂ ÂÂÂÂ ÂÂÂÂ ÂÂÂÂ 10,14

And Again…

2|_40,_56__
2|_20,_28__
2|_10,_14__
ÂÂÂÂ ÂÂÂÂ ÂÂÂÂ ÂÂÂÂ ÂÂÂÂ 5,7

Now, you can’t divide 5 and 7 anymore with any common number except 1.

This means: stop, and multiply all the numbers on the sides.
Here, that means multiply 2 * 2 * 2 = 8

That means that the HCF is 8.
To find the LCM, multiply the numbers on the side and the numbers at the very bottom.
Here that means 2 * 2 * 2 * 5 * 7
4*2*5*7
8*5*7
40*7
280

This means that the L.C.M. is 280!

That’s All!

If you try these methods and read this a couple of times, you should get the hang of HCF and LCM. Maybe you can even try finding the HCF and LCM of 3 or more numbers…!

Hope you enjoyed and learned!!!