The factors of 105 are the numbers that divide 105 exactly without leaving any remainder. The factors of 105 can be positive or negative. Similarly, the pair factors of 105 can also be in positive or negative form. For example, the pair factors of 105 can be (1, 105) or (-1, -105). If we multiply a pair of negative factors, such as multiplying -1 and -105, it will result in the original number 105. In this article, we are going to discuss the factors of 105, pair factors and the prime factors 105 using the prime factorization method and many solved examples.
Table of Contents:
- What are the Factors of 105?
- Pair Factors of 105
- Factors of 105 by Division Method
- Prime Factorization of 105
- Solved Examples
- FAQs
What are the Factors of 105?
The numbers that divide 105 exactly and leave a remainder zero, then the numbers are the factors of 105. In other words, a pair of numbers, which are multiplied together resulting in 105 are the factors of 105. As 105 is a composite number, it has many factors other than one and the number itself. Hence, the factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105 and its negative factors are -1, -3, -5, -7, -15, -21, -35 and -105.
Factors of 105: 1, 3, 5, 7, 15, 21, 35 and 105. Prime Factorization of 105: 3 × 5 × 7 |
Pair Factors of 105
The pair factors of 105 are the pair of numbers, which on multiplication result in the original number 105. As discussed above, the pair factors of 105 can be positive or negative. Thus, the positive and negative pair factors of 105 are given below:
Positive Pair Factors of 105:
Positive Factors of 105 |
Positive Pair Factors of 105 |
1 × 105 |
(1, 105) |
3 × 35 |
(3, 35) |
5 × 21 |
(5, 21) |
7 × 15 |
(7, 15) |
Hence, the positive pair factors of 105 are (1, 105), (3, 35), (5, 21), and (7, 15).
Negative Pair Factors of 105:
Negative Factors of 105 |
Negative Pair Factors of 105 |
-1 × -105 |
(-1, -105) |
-3 × -35 |
(-3, -35) |
-5 × -21 |
(-5, -21) |
-7 × -15 |
(-7, -15) |
Factors of 105 by Division Method
In the division method, the factors of 105 are found by dividing 105 by different integers. If the integer number divides 105 exactly leaving a remainder 0, then those integers are the factors of 105. Now, let us start dividing 105 by 1 and continue with the different integers.
- 105/1 = 105 (Factor is 1 and Remainder is 0)
- 105/3 = 35 (Factor is 3 and Remainder is 0)
- 105/5 = 21 (Factor is 5 and Remainder is 0)
- 105/7 = 15 (Factor is 7 and Remainder is 0)
- 105/15 = 7 (Factor is 15 and Remainder is 0)
- 105/21 = 5 (Factor is 21 and Remainder is 0)
- 105/35 = 3 (Factor is 35 and Remainder is 0)
- 105/105 = 1 (Factor is 105 and Remainder is 0)
If we divide 105 by any numbers other than 1, 3, 5, 7, 15, 21, 35 and 105, it leaves a remainder of some value. Hence, the factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.
Prime Factorization of 105
In the prime factorization of 105, the number 105 is written as the product of its prime factors. Now, let us discuss how to find the prime factors of 105.
Take a pair factor of 105, say (1, 105)
Here, factor 1 is neither prime nor composite, and it cannot be split further.
Now, take the other factor 105, which is a composite number, which can be factored further into its prime factors.
Thus, 105 can be written as the product of 3 and 35.
Now, 3 is a prime number and 35 is a composite number. Again, split the number 35 into its prime factors.
Hence, 35 is written as the product of 5 and 7, and now both the numbers 5 and 7 are prime numbers.
Now, write all the numbers in the form of a product of its prime factors.
Thus, 105 is written as 3 × 5 × 7
Therefore, the prime factorization of 105 is 3 × 5 × 7.
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Solved Examples
Example 1:
Find the common factors of 105 and 104.
Solution:
The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.
The factors of 104 are 1, 2, 4, 8, 13, 26, 52 and 104.
Thus, the common factor of 105 and 104 is 1.
Example 2:
Find the common factors of 105 and 106.
Solution:
Factors of 105 = 1, 3, 5, 7, 15, 21, 35 and 105.
Factors of 106 = 1, 2, 53 and 106.
Hence, the common factor of 105 and 106 is 1.
Example 3:
Find the common factors of 105 and 103.
Solution:
The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.
The factors of 103 are 1 and 103.
As 103 is a prime number, the common factor of 105 and 103 is 1.
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Frequently Asked Questions on Factors of 105
What are the factors of 105?
The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.
What is the prime factorization of 105?
The prime factorization of 105 is 3 × 5 × 7.
What are the positive pair factors of 105?
The positive pair factors of 105 are (1, 105), (3, 35), (5, 21) and (7, 15).
Write down the negative pair factors of 105.
The negative pair factors of 105 are (-1, -105), (-3, -35), (-5, -21) and (-7, -15).
Is 21 a factor of 105?
Yes, 21 is a factor of 105, as it divides 105 exactly without leaving a remainder.