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03 Oct 2022, 02:53
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Factors – What is it?
Factors (also known as divisors) are those numbers that completely divide the original number.
Suppose we say that: ‘a’ is a factor of ‘b’, then what does this mean?
Let us take an example to understand this.
If we ask you, what are the factors of 10? What will be your response?
Example 2:
Similarly, if we ask you, what are the factors of 24? What will be your response?
Number of factors – Easy to find?
Now that we have understood what are ‘factors’, let’s move to another important question, “How do we calculate the number of factors for a number?”
But what if we ask you, “How many factors does 120 have?”
General formula
To generalise the formula for the total number of factors,
If a number N can be prime factorised as
then
Let’s now answer the question we asked earlier. “How many factors does 120 have?”
Solution:
120 can be prime factorised as \(120=2^3×3^1×5^1\)
Thus, total number of factors of 120 \(=(3+1)×(1+1)×(1+1)=4×2×2=16 \)
Number of odd factors
We have discussed the rationale in this article. If you want to know more about this rule, you can go through this link.
We can generalize the formula for the total number of odd factors as below,
If a number N can be prime factorised as
What is the number of odd factors of 120?
Solution:
120 can be prime factorised as 120 \(=2^3×3^1×5^1\)
Number of even factors
Although there is a formula for this, we should not be worried about it at all.
Let’s take an example:
What is the number of even factors of 72?
Solution:
72 can be prime factorised as 72 \(=2^3×3^2\)
Before we finish off this article, let’s take one last example of 2700 and find its total number of factors, number of odd factors and number of even factors.
factors2700.png [ 18.84 KiB | Viewed 1840 times ]
Factors (also known as divisors) are those numbers that completely divide the original number.
Suppose we say that: ‘a’ is a factor of ‘b’, then what does this mean?
- It means that the number ‘a’ completely divides the number ‘b’ leaving no remainder.
- It can also be expressed as
\(\frac{b}{a}=K\) (where K is a positive integer)
Let us take an example to understand this.
If we ask you, what are the factors of 10? What will be your response?
- Since we know factors of a number are those numbers that divide the given number without leaving any remainders, factors of 10 will be 1, 2, 5 and 10.
- All these numbers i.e., 1, 2, 5 and 10 will leave no remainder when divided by 10.
\(10=1\times 10+0\)
\(10=2\times 5+0\)
\(10=5\times 2+0\)
\(10=10\times 1+0\)
Example 2:
Similarly, if we ask you, what are the factors of 24? What will be your response?
- We can use the same logic and say that the factors of 24 will be 1, 2, 3, 4, 6, 8, 12 and 24.
Number of factors – Easy to find?
Now that we have understood what are ‘factors’, let’s move to another important question, “How do we calculate the number of factors for a number?”
- For instance, in our earlier examples, we saw that
- 10 has 4 factors – 1. 2, 5 and 10
- 24 has 8 factors – 1, 2, 3, 4, 6, 8, 12 and 24
But what if we ask you, “How many factors does 120 have?”
- One way is to write down all the factors and count them manually.
- But as you must have already realised that it will be a tedious process
- So, is there any alternate way to find this out?
- Yes, there is one. We have discussed that in detail in our article using the example of 24. You can go through that if you wish to understand the logic behind the same.
General formula
To generalise the formula for the total number of factors,
If a number N can be prime factorised as
\(N=(P_1)^a\times (P_2)^b\times (P_3)^c ..... \)
(where, \(P_1,P_2,P_3\) and so on are distinct prime numbers)
then
total number of factors \(=(a+1)×(b+1)×(c+1)..... \)
Let’s now answer the question we asked earlier. “How many factors does 120 have?”
Solution:
120 can be prime factorised as \(120=2^3×3^1×5^1\)
Thus, total number of factors of 120 \(=(3+1)×(1+1)×(1+1)=4×2×2=16 \)
Number of odd factors
We have discussed the rationale in this article. If you want to know more about this rule, you can go through this link.
We can generalize the formula for the total number of odd factors as below,
If a number N can be prime factorised as
\(N=(2)^a\times (P_1)^b\times (P_2)^c ..... \)
(where \(P_1,P_2,P_3\) and so on are distinct prime numbers apart from 2)
thennumber of odd factors \(=(b+1)×(c+1).....\)
- The idea is to ignore the power of prime number 2 and treat the remaining number as a new number and find out the factors of it.
What is the number of odd factors of 120?
Solution:
120 can be prime factorised as 120 \(=2^3×3^1×5^1\)
- Now we will completely ignore the power of 2.
- Thus, the number will turn to \(3^1×5^1\)
- Now, the number of factors of this number \(= (1+1)×(1+1)=2×2=9\)
Number of even factors
Although there is a formula for this, we should not be worried about it at all.
- The number of factors can be either odd or even.
number of even factors = Total number of factors – Number of odd factors
Let’s take an example:
What is the number of even factors of 72?
Solution:
72 can be prime factorised as 72 \(=2^3×3^2\)
- Total number of factors \(=(3+1)×(2+1)=4×3=12\)
- Number of odd factors \(=(2+1)=3\)
Before we finish off this article, let’s take one last example of 2700 and find its total number of factors, number of odd factors and number of even factors.
Attachment:
factors2700.png [ 18.84 KiB | Viewed 1840 times ]
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block below for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
03 Oct 2022, 23:49
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06 Oct 2022, 07:48
Kudos
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A couple of questions are posted related to this article.
You can check the above post for the link to these questions. Solutions will be updated within 48 hours. Please have a try at these.
You can check the above post for the link to these questions. Solutions will be updated within 48 hours. Please have a try at these.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
