**Factors and Multiples:**

In advance level, you will learn this important topic of arithmetic in detail. But here, you’ll just learn what are these terms.

**Multiples:**

To understand this term, we need to know tables, i.e., tables of 2, tables of 3 etc.

For instance,

Table of 2:

2 × 0 = **0**

2 × 1 = **2**

2 × 2 = **4**

2 × 3 = **6**

2 × 4 = **8**

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You see that when 2 multiplies with an integer, it results to another integer (multiple), i.e., in the above table of 2, all resultant values (0, 2, 4, 6, 8, …) that are **boldfaced** and on the right side of equality sign are multiples of 2.

Similarly,

Table of 3:

3 × 0 = **0**

3 × 1 = **3**

3 × 2 = **6**

3 × 3 = **9**

3 × 4 = **12**

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Again you see that when 3 multiplies with an integer, it results to another integer. This resulted integer is called multiple of 3, i.e., in the above table of 3, all resultant values (0, 3, 6, 9, 12, …) that are **boldfaced** and on the right side of equality sign are multiples of 3.

**Factors:**

When an integer is divided by its factor, it always results to another integer. To understand this term, we need to remember the equation below:

For instance, from the above table of 2, when 6 is divided by 2, the result is 3, which is an integer. Therefore, we can say that 2 is a factor of 6.

In other words, from the tables above as,

2 × 4 = 8

It means 2 and 4 both are factors of 8, because, when each of them is divided by 8 results to one another, i.e an integer resulted.

**Important Note:** *When ‘a’ is a factor of ‘b’, then ‘b’ would always divisible by ‘a’. i.e., 3 is a factor of 12, so 12 is divisible by 3. In other words, a multiple is always divisible by its factor.*.

**Important Note:** *‘0’ is a multiple of every integer. And ‘1’ is a factor of every integer. But ‘0’ is not a factor of any integer. Because any integer divided by ‘0’ is not an integer, it’s undefined.*.

**Prime Factorization:**

The word *‘Prime’* itself indicates these numbers are of high power numbers like ‘Prime Minister’ (the highest minister in rank). All positive integers greater than 1 can be broken down to its primes.

For instance,

Therefore, prime factorization of 12 is:

12 = 2 × 2 × 3

⇒ **= 2 ^{2} × 3**

**Important Note:** *All integers greater than ‘1’ are multiples of some prime number*.

**Factors vs Prime Factors:**

Many students confuse with the two terms: *‘Factors’* and *‘Prime Factors’*. You should be clear about the two terms and never mix them.

Factor is a parent category, i.e., every integer has its factors, but all those factors which are prime numbers are its prime factors, e.g.,

**Factors** of 12: 1, 2, 3, 4, 6, and 12 itself. (As these integers, when divided by 12, results to an integer)

**Prime factors** of 12: 2, 2 and 3 (As we have already solved this before)

Also note,

**Unique Prime factors** of 12: 2 and 3.

**Important Note:** *All prime factors are also factors of an integer, but all factors are not necessarily prime factors of that integer. In other words, factors are the parent category of prime factors. So prime factors are sub-category or a part of factors.*

**Least Common Multiple (LCM):**

LCM, which is an abbreviation of Least Common Multiple, is mostly used in arithmetic calculations. So let’s learn how to take LCM at much quick time.

In your schooling era, perhaps you’ve first time studied this term. But that method is long one, because that method is for kids. From onward, you should remember the following way of taking L.C.M:

**Steps to take L.C.M:**

Suppose, we need to find value of (^{1}⁄_{12} + ^{1}⁄_{16}).

For that purpose, we need L.C.M of 12 and 16.

Write the two integers as follows:

12, 16

Now, think about the highest integer which can be taken as common from both 12 and 16.

(It might be easy in this case. You clearly know 4 can be taken as common from both 12 and 16. But in case of higher integers, you may take highest possible integer that you might think on first glance, and then from remaining integers, you may take common again until no integer can be taken as common from the two integers. And then simply multiply all those integers that were taken as common.)

To understand the above statement written in parenthesis, let’s take 2 common instead of 4 from the above two integers.

2 (^{12}⁄_{2}, ^{16}⁄_{2}) (Or you may simply skip this step)

2 (6, 8)

Now, remaining two numbers that were left are 6 and 8. Now think whether an integer greater than 1 still be taken as common from these remaining integers (i.e 6 and 8)?

Yes! 2 can again be taken as common from 6 and 8.

Hence,

2 × 2 (^{6}⁄_{2}, ^{8}⁄_{2}) (Again you may simply skip this step)

4 (3, 4)

Now, we cannot take any further common from the remaining integers inside parenthesis (i.e 3 and 4). Stop at this stage, and simply multiply all integers inside and outside parenthesis. i.e

4 × 3 × 4 = 12 × 4 **= 48**

So,

**Taking L.C.M by skipping unimportant steps:**

In above method, several steps are taken so that people from medical or arts background can easily learn. After getting this, you’ll able to solve in just few steps as below:

Suppose we need to find L.C.M of 24 and 36

As we know, 12 can be taken as common from the above two integers, so

12 (2, 3)

Thus,

L.C.M of 24 and 36 = 12 × 2 × 3 = 12 × 6 **= 72** **Answer**

You may do this even more fast after adopting this method and do some practice.

**L.C.M through Multiples:**

Let’s find the L.C.M of 6 & 8.

The L.C.M of 6 and 8 gives us the integer that is least common in multiples of both 6 and 8. i.e.,

We can see that 24 is the least common in multiples of both 6 and 8. So the L.C.M of 6 & 8 is 24.

**L.C.M through Prime factorization:**

Let’s have a look below,

First we need to make prime factorization of these two. i.e.,

Now, write down it in an order as figure bellow. Find the common prime factors in prime factorization of 6 & 8. Count this common factor only once rather than twice in L.C.M, while write down remaining primes which are not common.

Finally, multiply all the primes that were brought down. It would give us the L.C.M

So L.C.M of 6 & 8 is:

L.C.M = 2 × 3 × 2 × 2 = 6 × 4

⇒ **= 24** **Answer**

Remember that when L.C.M of two integers are required such that one of the integer is the multiple of the other integer, the L.C.M would be that integer which is multiple. For instance,

L.C.M of 6 and 18 is always 18, because 18 is multiple of 6. So here we don’t need to find L.C.M as we already have.

Also remember that, when the L.C.M of more than two integers are required, just start solving by considering any two integers and then consider other integers one-by-one. For instance,

Suppose, we need to find L.C.M of *a*, *b*, and *c*.

First take find L.C.M of *a* and *b* (suppose L.C.M is *x* here). After that take L.C.M of *x* and *c*. Thereby you’ll get the final L.C.M of *a*, *b*, and *c*.

In case of four or more integers, i.e

*a*, *b*, *c* and *d*

First take find L.C.M of *a* and *b*, suppose it’s *x*

Then, find L.C.M of *c* and *d*, suppose it’s *y*

Finally, you may find L.C.M of *x* and *y*, that will be our required L.C.M of *a*, *b*, *c* and *d*

**Important note in L.C.M:**

As I said, keep an eye on multiples. i.e if you see any of the integer to be a multiple of other, just ignore the factor. For instance,

Suppose we need to find L.C.M of *a*, *b*, *c* and *d*. And you analyze that *c* is actually multiple of *a*; then just ignore *a*, and find L.C.M of *b*, *c* and *d*.

**Greatest Common Factor (GCF) or Highest Common Factor (HCF):**

GCF or HCF also involves two or more integers, from which we need to find their G.C.F; e.g, let’s find the G.C.F of 6 & 8.

The G.C.F of 6 and 8 gives an integer that is greatest factor in common of 6 & 8. i.e.,

We can see that 2 is the greatest common in factors of both 6 and 8. So the G.C.F of 6 & 8 is 2.

Let’s have a look below,

First we need to make prime factorization of these two. i.e.,

Now, write down it in an order as figure below. Find the common prime factors in prime factorization of 6 & 8. Count this common factor only once rather than twice in G.C.F. But don’t write down the remaining primes that are not in common in prime factorization of both 6 & 8.

Therefore, the G.C.F of 6 & 8 is **2**.

**Rationalization:**

When we solve the following expression for its simplification, this process is known as rationalization, i.e.,

As we can multiply as well as divide an expression with same number. So by multiplying and dividing the above expression with √2 + 1 we’ll get,

This process of transferring square root from denominator to numerator, or simply eliminating the square root from the denominator is known as rationalization.

**Application of rationalization concept:**

Rationalization is used in simplifying expressions such as:

I’m sure this application has clear your understanding about rationalization concept importance in arithmetic.

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