 # Factor of 105

### What is the Factor of 105?

Two numbers when multiplied together and obtain the result as 105, then these numbers will be known as factors of 105.

Pair factors of 105 can be both positive and negative but not in decimal or fraction.  Let us understand the concept through an example:

The factor pair of 105 is (1, 105) and (-1,-105)

Because, when we multiply the factors 1 and 105 we get original number 105

Similarly, if we multiply the two negative factors -1 and -105 we get the original number 105.

Therefore, we can say both positive and negative pair factors of 105 are the factors of 105.

Now, we will use the similar factorization method to find the multiples and factors of 105. In this particular method, we will first examine the number itself and 1 as the factor. So 1 and 105 are the factors of 105 and we will continue finding the factors of 105 which will give us the original number 105.

### Pair Factors of 105

We will find the pair factors of 105 by multiplying the two numbers in a pair that will eventually give the original number 105.

Positive Pair Factors of 105 are

(1,105)

(3, 35)

(5, 21)

(7, 15)

(15, 7)

(21, 5)

(35, 3)

(105, 1)

Negative Pair Factors of 105

(-1,-105)

(-3, -35)

(-5, -21)

(-7, -15)

(-15, -7)

(-21, -5)

(-35, -3)

(-105, -1)

### All Factors of 105 in Pairs

Here, you can see  all factors of 105 in pairs

(1,105) are factors of 105 as 1 x 105 is 105

(3, 35) are factors of 105 as 3 x 35 is 105

(5, 21) are factors of 105 as 5 x 21 is 105

(7, 15) are factors of 105 as 7 x 15 is 105

### Factor Tree of 105

One method to find the prime factorization of a number is to make a factor tree. In factor trees, the factors of numbers are first recognised and then those numbers are further factorised until we reach closure.

The find step of constructing a factor tree is to find the pairs of a factor whose product of the numbers we are factoring. These two factors are the first branch of the factor tree. There are generally various pairs of factor factors that we chose to initiate the process. We repeat the process with each factor until every branch of the tree ends in a prime number.

Here, you can see the factor tree of 105

### Solved Examples

1. Find the prime factorization of 700 and 100 and then find the prime factorization of 7000 knowing that 7000 = 100 x 70.

Solution: Prime factorization of 700 and 100 are

All prime factors of 700 = 2 x 5 x 7

All prime factors of 100 = 2 x 2 x 5 x 5

Knowing the fact that 7000 = 100 x 70 to find the prime factorization of 7000.

7000 = 100 x 70 = 22 x 52 x (2 x 5 x7) = 2 x 2 x 5 x 5 x 2 x 5 x 7 = 23 x 53 x 7

2.  Find the common factor of 6 and 8

All factors of 6 = 1, 2, 3 and 6

All factors of 8 = 1, 2, 4 and 8

Hence, the common factors of 6 and 8 are 1 and 2.

3.  What is the prime factorization of the following numbers?

28, 32, and 100

Prime factors of 28 are = 2 x 2 x 7 = 22 x 7

Prime factors of 100 are = 2 x 2 x 5 x 5 = 22 x 55

Prime factors of 32 are = 2 x 2 x 2 x 2 x 2 = 25

### Fun Facts

• All the natural numbers are a minimum one pair factor.

• 1 is the factor of each and every number. For example, 1 x 6= 6, 1x 13= 13, 1 x 63 = 63

• In division, divisor and quotient are both considered as the factors of the dividend if the remainder is 0.

### Quiz Time

1. Which of the below is the factor of 72?

1.  8 and 9

2. 12 and 7

3. 36 and 36

4. 9 and 7

5. None of these

2. How many pair factors are there for the number 28?

1. 1

2. 2

3. 3

4. 4

3. The common factors of 56 and 44 are

1. 1,2,4

2. 1,2,11

3. 1,8

4. 2

1. What is the Prime Factorization of 105?

Prime factorization is the method of finding the prime factors that can be multiplied together to make the original number. For example prime factors of 105 are 3, 5 , 7 as 3 x 5 x 7 = 105 and 3,5 and 7 are prime numbers.

Prime factorization of 105 = 3¹ x 5¹ x 7¹

The prime numbers when multiplied by any whole numbers or natural numbers but not by 0 are known as the composite numbers. Generally, prime factorization is performed on the composite numbers to find the prime factors. This method is also used to find the least common multiple and highest common factors. If two numbers are given, then the highest common factor is the largest factor included in both the numbers whereas the least common multiple is the smallest common multiple of both the numbers.

2. Explain the Common Factors.

In Mathematics, common factors are defined as the factors that are common to two or more numbers.

To find the common factors, the first step is to write down all the factors of two numbers separately and then compare them. Now, list all the factors that are common to both the numbers. These factors are known as common factors of given two numbers.

For examples, common factors of 105 and 100 are

Factors of 105 = 3 x 5 x 7

Factors of 100 = 2  x 2 x 5 x 5

Hence, the common factors of 105 and 100 are 5 as 5 are the common number between 100 and 105.