Introduction To HCF

The term $HCF$ means $Highest\ Common\ Factor$, this is also known as $GCD$ or $Greatest\ Common\ Divisor$.

When two or more numbers share a factor, then it is a common factor between the two numbers.

Let us take the number $12$ and $18$.

Both of them are divisible by $2$, so, $2$ is a common factor between them.

Both of them are also divisible by $3$, so $3$ is also a common factor between them.

The largest factor that can divide both the numbers is called the $Highest\ Common\ Factor$

One of the easy ways to find the $HCF$ of two numbers is to factorize both the numbers into prime factors and see how many factors are common between the two numbers.

We will take an example to understand this.

Let us find out the $HCF$ of $24$ and $36$

Factorising $24$ we get:

$24 = 2 \times 2 \times 2 \times 3$

And factorising $36$ we get:

$36 = 2 \times 2 \times 3 \times 3$

Now we can see that there are two $2$s and one $3$ common between both the numbers. Therefore, the $HCF$ of $24$ and $36$ is $2 \times 2 \times 3 = 12$

We will take one more example to understand this better. We will take the one more example to understand this.

Let us find the $HCF$ of $90$ and $135$.

If we factorize $90$, we get:

$90 = 2 \times 3 \times 3 \times 5$

$135 = 3 \times 3 \times 3 \times 5$

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We can see that we have two $3$s and one $5$ common in both the number.

Therefore, the $HCF$ of $135$ and $90$ is $3 \times 3 \times 5 = 45$

One important thing that we should remember is that the $HCF$ of any two different numbers cannot be greater than the difference of the two numbers. This is how we can understand it.

Let us say we have two different numbers $A$ and $B$, which have an $HCF$ of $g$

This means that $A$ is a multiple of $g$ and so is $B$.

Let us look at the number line below:

Since $A$ and $B$ are both multiples of $g$, both will fall at an exact number of leaps of length $g$. The number line shows some of the correct positions of $B$ with the $\unicode{0x2713}$ sign and some incorrect position using the $\unicode{0x2717}$ sign. We can see the $B$ has to be at least $g$ length away from $A$.

It can also be any number of leaps of $g$ away but always a whole number of leaps. It can never be a point anywhere between two leaps.

That is why the $HCF$ of two numbers can never exceed their difference. You should also be able to observe that the difference of the two numbers should always be divisible by their $HCF$.