**Introduction**

Multiplication and division are two important operations in mathematics. We can multiply a number by any number. Similarly, a number can be divided by any number. Two important terms that are related to the multiplication and division of numbers are factors and multiples. Let us learn more about factors of 3.

**Definition**

**A factor of a number is an exact divisor of that number. **In other words, a factor of a number is that number that completely divides the number without leaving a remainder. Therefore, factors of 3 would be those numbers that exactly divide the number 3. Let us learn more about these numbers which are factors of 3.

**What are the factors of 3?**

The factors of the number 3 are 1, and 3. We can see that there are only 2 factors of the number 3. This means that there are only 2 numbers that divide the number 3 completely without leaving any remainder.

**How to verify factors of 3?**

Let us now verify the factors of 3.

We shall use the division method for finding the factor of 3

**Division by 1**

Let us first divide 3 by 1. We know that every number is divisible by 1. So, 1 is a factor of 3. . . . . . . . . . . . . . . . . . . . . ( 1 )

**Division by 2**

Let us divide 3 by 2. We will have,

3 ÷ 2 = 1 with remainder 1

This means that 3 is not completely divided by 2 giving a quotient 1 and remainder 1, hence 2 is not a factor 3. . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

So, we have 2 x 1 + 1 = 3

Now let us find the factors of 3

**Division of 3**

We know that 3 is a prime number. This means that there are only two factors of 3, namely 1 and the number 3 itself. Hence, it cannot be further divided into smaller factors. This marks the end of the division for finding the factors of 3. . . . . . . . . . . . . . . . . . . . ( 3 )

From ( 1 ), ( 2 ) and ( 3 ), we have,

**The factors of 3 are 1 and 3**

**Using the Divisibility rule for finding the factors of 3**

We know that we have a defined set of divisibility rules that allow checking to check whether a number is a factor of another number or not. Let us use these divisibility rules to verify the factors of 3.

Divisible by | Divisibility Rule | Is 3 divisible by this number? | Reason |

2 | If a number is even or a number whose last digit is an even number i.e. 2,4,6,8 including 0, it is always completely divisible by 2. | No | 3 is not an even number |

3 | The divisibility rule for 3 states that a number is completely divisible by 3 if the sum of its digits is divisible by 3. | Yes | Every number is divisible by itself |

**Pairing the Factors of 3**

Let us now pair the factors of 3. We know that the factors of 3 are 1 and 3. Now, let us pair these numbers. We know that 1 x 3 = 3. This means that there can be only one pair of factors of 3. This only pair is ( 1 and 3 ). This pair of factors of 3 can thus be represented in the below table as –

Factor of 3 | Pairs of factors of 3 |

1 and 3 | Only pair – 1 and 3 since 1 x 3 = 3 |

If we wish to find out a negative pair as factors of 3, it would be -1 and -3 as

**( – 1 ) x ( – 3 ) = 3 **

Now that we know the factors of 3 let us analyse what type of numbers are the factors 3

**Factors of 3 by Division Method**

We can find the factors of by dividing the number 3 by integer numbers. It is important to note here that there are only three positive integers that are less than or equal to 3, i.e. the numbers 1, 2 and 3 are the only three positive integers that are less than or equal to 3. So let us divide the number 3 by these three numbers one by one. We will have,

- 3 when divided by 1 will give 3 as quotient and 0 as remainder. Hence 3 is divisible by 1
- 3 when divided by 2 will give 1 as quotient and 1 as remainder. Hence 3 is not divisible by 2
- 3 when divided by 3 will give 1 as quotient and 0 as remainder. Hence 3 is divisible by 3

From the above statement we can say that there are only positive integers that divide 3 completely and these numbers are 1 and the number 3 itself.

**Factors of 3 by Prime Factorisation**

We know that a number having only two factors is called a **prime number**. The two factors are the number 1 and the number itself. For example, consider the number 7. The number 7 has only factors, 1 and the number 7 itself. Therefore, 7 is a prime number. Similarly, the number 3 is also a prime number as it has only two factors, 1 and the number 3 itself. So, does the number 3 have any prime numbers as its factor? Let us find out.

Let us list down the factors of 3. The factors are 1, and 3. Let us analyse each factor one by one.

**1 as a factor of 3** – We know that the number 1 is neither prime nor composite. Hence though 1 is a factor of 3, it is not a prime number. So, we can say that 1 is not a prime factor of 3.

**3 as a factor of 3** – We know that 3 is a prime number. This is because it has only two factors, 1 and the number 3 itself. Hence, 3 is a prime factor of itself, i.e. 3.

From the above discussion, we can say that the prime factors of 3 are –

Prime Numbers as Factors of 3 | 3 |

Hence, there is one prime factor of 3.

**Key Facts and Summary**

- A factor of a number is an exact divisor of that number. In other words, a factor of a number is that number that completely divides the number without leaving a remainder. The factors of the number 3 are 1 and 3.
- If we wish to find out a negative pair as factors of 3, it would be -1 and -3 as ( – 1 ) x ( – 3 ) = 3

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