The two numbers which get multiplied and produce the result as 415, then these numbers are the factors of 415. These numbers can be positive or negative but cannot be in decimal form or fractions. Let’s take an example to understand this concept. The factor pair for number 415 will be (1, 415) (-1, -415). When we multiply the factors (1, 415) we get the number 415. Similarly, even when we multiply the two negative factors(-1, -415) then we get the number 415 only. In this article, we will learn the pair factors and the prime factors of 415 using the prime factorization method and many solved examples.
Table of Contents:
- What are the Factors of 415?
- Pair Factors of 415
- Factors of 415 by Division Method
- Prime Factorization of 415
- Solved Examples
- FAQs
What are the Factors of 415?
The numbers that divide 415 exactly without leaving the remainder are the factors of 415. In other words, the factors of 415 are the numbers that are multiplied in pairs resulting in an original number 415. As the number 415 is a composite number, it has many factors other than 1 and 415. Thus, the factors of 415 are 1, 5, 83 and 415. Similarly, the negative factors of 415 are -1, -5, -83 and -415.
Factors of 415: 1, 5, 83 and 415 Prime Factorization of 415: 5 × 83 |
Pair Factors of 415
The pair factors of 415 are the numbers that are multiplied in pairs that result in an original number 415. As discussed above, the pair factors of 415 can be positive as well as negative, but they cannot be in decimal or fractional form. Thus, the positive and negative pair factors of 415 are given below:
Positive Pair Factors of 415:
Positive Factors of 415 | Positive Pair Factors of 415 |
1 × 415 | (1, 415) |
5 × 83 | (5, 83) |
Negative Pair Factors of 415:
Negative Factors of 415 | Negative Pair Factors of 415 |
-1 × -415 | (-1, -415) |
-5 × -83 | (-5, -83) |
Factors of 415 by Division Method
The factors of 415 can be found using the division method. In the division method, the number 415 is divided by different consecutive integers. If the integer divides 415 exactly and leaves the remainder 0, then the integers are the factors of 415. Now, let us start dividing 415 by 1 and continue with the consecutive integers.
- 415/1 = 415 (Factor is 1 and remainder is 0)
- 415/5 = 83 (Factor is 5 and remainder is 0)
- 415/83 = 5 (Factor is 83 and remainder is 0)
- 451/415 = 1 (Factor is 415 and remainder is 0)
If we divide 415 by numbers other than 1, 5, 83 and 415, we will get the remainder. Thus, the factors of 415 are 1, 5, 83 and 415.
Prime Factorization of 415
The process of writing the numbers as the product of the prime factors of 415 is called the prime factorization of 415. Now, let us learn how to find the prime factorization of 415.
Take, the pair factor of 415, say (1, 415)
We know that the number 1 is neither prime nor composite. So, leave the number 1 as it is.
Now, take the other number in the pair factor 415.
As 415 is a composite number, it can be further split into its prime factors. 415 can be factorized as the product of 5 and 83.
Now, both the numbers 5 and 83 are the prime factors.
Thus, the prime factorization of 415 can be written as 1× 5 × 83 or 5^{1} × 83^{1}.
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Examples
Example 1:
Find the common factors of 415 and 414.
Solution:
The factors of 415 are 1, 5, 83 and 415.
The factors of 414 are 1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414
The common factor of 415 and 414 is 1.
Example 2:
Find the common factors of 415 and 416.
Solution:
Factors of 415 = 1, 5, 83 and 415.
Factors of 416 = 1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416
Thus, the common factor of 415 and 416 is 1.
Example 3:
Find the common factors of 415 and 420.
Solution:
The factors of 415 are 1, 5, 83 and 415.
The factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420
Thus, the common factors of 415 and 420 are 1 and 5.
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Frequently Asked Questions on Factors of 415
What are the factors of 415?
The factors of 415 are the numbers that divide 415 exactly without leaving a remainder. Thus, the factors of 415 are 1, 5, 83 and 415.
What is the prime factorization of 415?
The prime factorization of 415 is 5 × 83.
What are the positive pair factors of 415?
The positive pair factors of 415 are (1, 415) and (5, 83).
Write down the negative pair factors of 415.
The negative pair factors of 415 are (-1, -415) and (-5, -83)
Is 5 a factor of 415?
Yes, 5 is a factor of 415. As 5 divides 415 exactly without leaving a remainder, 5 is a factor of 415.