**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 11.

**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 11 would be those numbers that exactly divide the number 11. Let us learn more about these numbers which are factors of 11.

**What are the factors of 11?**

The factors of the number 11 are – 1, and 11. We can see that there are only 2 factors of the number 11. This means that there are only 2 numbers that divide the number 11 completely without leaving any remainder.

**How to verify factors of 11?**

Let us now verify the factors of 11.

We shall use the division method for finding the factor of 11

**Division by 1**

Let us first divide 11 by 1. We know that every number is divisible by 1. So, 1 is a factor of 11. . . . . . . . . . . . . . . . . . . . . ( 1 )

**Division by 2**

Let us divide 11 by 2. We will have,

11 ÷ 2 = 5 with remainder 1

This means that 11 is not completely divided by 2 giving a quotient 5 and remainder 1, hence 2 is not a factor 11. . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

So, we have 2 x 5 + 1 = 11

Now let us find the factors of 11

**Division of 11**

We know that 11 is a prime number. This means that there are only two factors of 11, namely 1 and the number 11 itself. Hence, it cannot be further divided into smaller factors. This marks the end of the division for finding the factors of 11. . . . . . . . . . . . . . . . . . . . ( 3 )

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

**The factors of 11 are 1 and 11**

**Using the Divisibility rule for finding the factors of 11**

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 11.

Divisible by | Divisibility Rule | Is 11 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 | 11 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. | No | 1 + 1 = 2 and 2 is not divisible by 3. |

4 | If the last two digits of a number are divisible by 4, then that number is a multiple of 4 and is divisible by 4 completely. | No | 11 only 2 digits, and 2 is not divisible by 4. |

5 | Numbers, which last with digits, 0 or 5 are always divisible by 5. | No | 11 ends with 1, not 0 nor 5. |

6 | Numbers which are divisible by both 2 and 3 are divisible by 6. That is, if the last digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6. | No | 11 is neither divisible by 2 nor by 3. |

7 | The divisibility rule of 7 states that if we get a difference that is also divisible by 7 by subtracting the last digit’s number and 2’s product, then the number is divisible by 7 | No | 11 is not divisible by 7 |

8 | If the last three digits of a number are divisible by 8, then the number is completely divisible by 8. | No | 11 is not divisible by 8 |

9 | The rule for divisibility by 9 is similar to the divisibility rule by 3. That is, if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9. | No | 1 + 1 = 2 and 2 is not divisible by 9. |

10 | The divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10. | No | 11 does not end with 0. |

**Pairing the Factors of 11**

Let us now pair the factors of 11. We know that the factors of 11 are 1 and 11. Now, let us pair these numbers. We know that 1 x 11 = 11. This means that there can be only one pair of factors of 11. This only pair is ( 1 and 11 ). This pair of factors of 11 can thus be represented in the below table as –

Factor of 11 | Pairs of factors of 11 |

1 and 11 | Only pair – 1 and 11 since 1 x 11 = 11 |

Now that we know the factors of 11 let us analyse what type of numbers are the factors 11

**Prime Factors of 11**

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 11 is also a prime number as it has only two factors, 1 and the number 11 itself. So, does the number 11 have any prime numbers as its factor? Let us find out.

Let us list down the factors of 11. The factors are – 1, and 11. Let us analyse each factor one by one.

**1 as a factor of 11** – We know that the number 1 is neither prime nor composite. Hence though 1 is a factor of 11, it is not a prime number. So, we can say that 1 is not a prime factor of 11.

**11 as a factor of 11** – We know that 11 is a prime number. This is because it has only two factors, 1 and the number 11 itself. Hence, 11 is a prime factor of itself, i.e. 11.

From the above discussion, we can say that the prime factors of 11 are –

Prime Numbers as Factors of 11 | 11 |

Hence, there is one prime factor of 11.

**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 11 are 1 and 11.

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