**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 65.

**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 65 would be those numbers that exactly divide the number 65. Let us learn more about these numbers which are factors of 65.

**What are the factors of 65?**

The factors of the number 65 are – 1, 5, 13 and 65. We can see that there are only 4 factors of the number 65. This means that there are only 4 numbers that divide the number 65 completely without leaving any remainder.

**How to verify factors of 65?**

Let us now verify the factors of 65.

We shall use the division method for finding the factor of 65

**Division by 1**

Let us first divide 65 by 1. We know that every number is divisible by 1. So, 1 is a factor of 65. . . . . . . . . . . . . . . . . . . . . ( 1 )

**Division by 5**

Let us divide 65 by 5. We will have,

65 ÷ 5 = 13

This means that 65 is completely divided by 5 giving a quotient 13 and remainder 0 hence 5 is a factor 13. . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

So, we have 5 x 13 = 65

Now let us find the factors of 13

**Division of 13**

We know that 13 is a prime number. This means that there are only two factors of 13, namely 1 and the number 13 itself. Hence, it cannot be further divided into smaller factors. This marks the end of the division for finding the factors of 65. . . . . . . . . . . . . . . . . . . . ( 3 )

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

**The factors of 65 are 1, 5, 13 and 65**

**Using the Divisibility rule for finding the factors of 65**

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 65.

Divisible by | Divisibility Rule | Is 65 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 | 65 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 | 6 + 5 = 11 and 11 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 | 65 only 2 digits which are not divisible by 4, and hence, 65 is not divisible by 4. |

5 | Numbers, which last with digits, 0 or 5 are always divisible by 5. | Yes | 65 ends 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 | 65 is not divisible by 2 or 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 | 65’s last product is 13 which 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 | 65 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 | 6 + 5 = 11 and 11 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 | 65 does not end with 0. |

**Pairing the Factors of 65**

Let us now pair the factors of 65. We know that the factors of 65 are 1, 5, 13 and 65. Now, let us pair these numbers. We know that 1 x 65 = 65 and 5 x 13 = 65. This means that there can be two pairs of factors of 65. These two pairs are ( 1 and 65 ) and ( 5 and 13 ). These pairs of factors of 65 can thus be represented in the below table as –

Factor of 65 | Pairs of factors of 65 |

1, 5, 13 and 65 | First pair – 1 and 65 since 1 x 65 = 65 |

Second Pair – 5 and 13 since 5 x 13 = 65 |

Now that we know the factors of 65, let us analyse what type of numbers are the factors 65

**Prime Factors of 65**

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 65 have any prime numbers as its factor? Let us find out.

Let us list down the factors of 65. The factors are – 1, 5, 13 and 65. Let us analyse each factor one by one.

**1 as a factor of 65** – We know that the number 1 is neither prime nor composite. Hence though 1 is a factor of 65, it is not a prime number. So, we can say that 1 is not a prime factor of 65.

**5 as a factor of 65** – We know that 5 is a prime number. This is because it has only two factors, 1 and the number 5 itself. Hence, 5 is a prime factor of 65.

**13 as a factor of 65** – We know that 13 is a prime number. This is because it has only two factors, 1 and the number 13 itself. Hence, 13 is a prime factor of 65.

**65 as a factor of 65** – We know that 65 is a composite number. This is because we already know that there are more than 2 factors of 65, namely, 1, 5, 13 and the number 65 itself. Hence, 65 is a factor of 65 but is not a prime number.

From the above discussion, we can say that the prime factors of 65 are –

Prime Numbers as Factors of 65 | 5 and 13 |

Hence, there are 2 prime factors of 65.

**Co-Prime Numbers as Factors of 65**

Let us recall that co-prime numbers are the numbers that have no common factor apart from 1. Co-prime numbers are different from prime numbers. While prime numbers are all odd numbers with the only exception being 2 which is an even number, the co-prime numbers can be both prime and composite with the only condition being that they have no other factor other than 1. Let us now check whether any co-prime numbers are the factors of 65.

We now know that the numbers 1, 5, 13 and 65 are the factors of 65. Let us consider these numbers as pairs one by one.

**1 and 5** – We know that 1 is neither a prime nor a composite number. 5 on the other hand is a prime number. Hence, the pair of 1 and 5 cannot be co-prime factors of 65.

**1 and 13** – We know that 1 is neither a prime nor a composite number. 13 on the other hand is a prime number. Hence, the pair of 1 and 13 cannot be co-prime factors of 65.

**1 and 65** – We know that 1 is neither a prime nor a composite number. 65 on the other hand is a composite number. Hence, the pair of 1 and 65 cannot be co-prime factors of 65.

**5 and 13** – We know that 5 is a prime number. The other number of the pair, 13 is also a prime number. This means that the numbers 5 and 13 have no common factors other than 1. Hence, we can say that **the pair of 5 and 13 is a pair of co-primes that is a factor of the number 65. **

**5 and 65** – We know that 5 is a prime number. 65 on the other hand is a composite number. Hence, the pair of 5 and 65 cannot be co-prime factors of 65.

**13 and 65** – We know that 13 is a prime number. 65 on the other hand is a composite number. Hence, the pair of 13 and 65 cannot be co-prime factors of 65.

**From the above discussion, we can say that the pair of 5 and 13 is a pair of co-primes that is a factor of the number 65. **

**Properties of factors of 65**

There are some general properties of factors that are satisfied by the factors of any number. The following general properties of factors are satisfied by the factors of 65 –

- 1 is a factor of every number. Factors of 65 satisfy this property as 1 is a factor of 65.
- Every number is a factor of itself. Factors of 65 satisfy this property as 65 is a factor of itself.
- Every factor of a number is an exact divisor of the number itself. Factors of 65 satisfy this property as 65 is an exact divisor of itself.
- Every factor of a number is less than or equal to that number. Factors of 65 satisfy this property as all factors of 65 are less than or equal to the number 65.
- Factors of a given number are finite. Factors of 65 satisfy this property as the number of factors of 65 is finite.

**Common Factors of 65 with other numbers**

We know that each number has its own set of factors. However, there are many numbers that have some factors in common. Those factors which are common among the factors of two or more numbers are known as common factors. Let us check whether 65 has a common factor with any other number.

We know that the factors of 65 are – 1, 5, 13 and 65

Let us consider another number say 45. What will be the factors of 45? Let us find out.

The factors of 45 will be 1, 3, 5, 9, 15 and 45

Can we see any common factors between 45 and 65? The answer is yes. The common factors are 1 and 5.

So, 1 and 5 are two common factors between 45 and 65. Out of these 5 is the highest common factor between these two numbers. Can we assign a name to this highest common factor? Yes. Let us learn more about it.

**The highest common factor ( H. C. F. ) of two or more numbers is the greatest or the largest among common factors. In other words, the H. C. F. of two or more numbers is the largest number that divides all the given numbers exactly. **

Let us understand how to find HCF between 65 and other numbers.

**Prime Factorisation method**

In order to find the highest common factor ( H. C. F. ) of two or more numbers, the following steps are followed-

- Obtain the numbers.
- Write the prime factorisation of all the numbers.
- Identify the common factors.
- For each common prime factor find the minimum number of times it occurs in the prime factorisation of the given numbers.
- Multiply each common prime factor the number of times determine the previous step and find their product to get the highest common factor ( H. C. F. ) of the given numbers.

Let us understand the above steps through an example.

Suppose we want to find the highest common factor ( H. C. F. ) of 45 and 65. We shall follow the above-listed steps for this purpose.

- Obtain the numbers. We have the numbers as 45 and 65.
- Write the prime factorisation of all the numbers. We have,

45 = 3 x 3 x 5 and 65 = 5 x 13 - Identify the common factors. We can see from above that the common factors of 45 and 65 is 5
- For each common prime factor find the minimum number of times it occurs in the prime factorisation of the given numbers. We can see that the common factor 5 appears one time in the prime factorisation of 45 while it appears one time in the prime factorisation of 65.
- Multiply each common prime factor the number of times determine the previous step and find their product to get the highest common factor ( H. C. F. ) of the given numbers. H. C. F. of 45 and 65 = 5 x 1 = 5

In this manner, we can find the highest common factor between 65 and other numbers.

**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 65 are – 1, 5, 13 and 65
- 1 is a factor of every number. Factors of 65 satisfy this property as 1 is a factor of 65.
- Every number is a factor of itself. Factors of 65 satisfy this property as 65 is a factor of itself.
- Every factor of a number is an exact divisor of the number itself. Factors of 65 satisfy this property as 65 is an exact divisor of itself.
- Every factor of a number is less than or equal to that number. Factors of 65 satisfy this property as all factors of 65 are less than or equal to the number 65.
- Factors of a given number are finite. Factors of 65 satisfy this property as the number of factors of 65 is finite.
- The pair of 5 and 13 is a pair of co-primes that is a factor of the number 65.
- The highest common factor ( H. C. F. ) of two or more numbers is the greatest or the largest among common factors. In other words, the H. C. F. of two or more numbers is the largest number that divides all the given numbers exactly.

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