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Factors of 3: Prime Factorization, Methods, Tree, and Examples
Factors of 3 are mathematical expressions that divide the number ultimately, without leaving any remains behind when undergoing division. In other words, factors of the given number 3 are also referred to as their divisors.
Figure 1 – All possible Factors of 3
Factors of 3 are also referred to as the integers that produce 3 as the result of their product. By product, we mean the result of multiplying the factors of 3 with one another such that these pairs are also termed the factor pairs of 3.
The number 3 is a perfect odd prime number. Due to its exceptional nature, it is clear that the number 3 is lucky to have just two main factors for itself, one of which will be the universal factor, i.e., 1, while the other one will be the original number 3 itself.
Interestingly, the number 3 can have positive as well as negative factors. The positive and negative factors of number 3 can be characterized as values with opposite signs. The negative factors of 3 are the numbers with a negative sign.
In this article, we will find the methods and techniques used to calculate the factors of the number 3, its prime factorization, factor tree, and pairs of factors.
What Are the Factors of 3?
Factors of 3 are 1 and 3. Both numbers result in a perfect-whole number quotient and a zero remainder when divided.
In other words, the total number of factors of number 3 is 2, with 3 being the largest factor.
How To Calculate the Factors of 3?
You can calculate the factors of 3 simply by using the universally used multiplication or division methods as one of the primary techniques. Let’s first look at the division process.
The division approach is one of the most common and straightforward techniques for calculating the factors. The best way to tell if the division technique worked correctly is to see if the divisor produced a whole-number quotient and a remainder of zero.
The divisor and the whole-number quotient are considered the factors of the given number.
Since 3 is the given number in this case. To determine the factors of 3, let’s first create a list of all the numbers that are less than or equal to 3.
Such that the list of the recommended numbers includes: 1, 2, and 3, respectively.
At first, divide 3 by the smallest recommended number, i.e., 1.
Check for the remainder.
\[ \dfrac {3}{1} = 3, r=0 \]
If the remainder obtained is zero, then the divisor is the factor of the number. Hence, in this case, the remainder is zero, so 1 is the factor of 3.
Also, as the above division process generated a perfect whole-number quotient, i.e., 3, three are introduced in the well-defined list of factors of 3.
Now, divide 3 by the number 2 such that,
\[ \dfrac {3}{2} = 1.5 \]
The number 1.5 is not considered the whole-number quotient of the above division. Also, the remainder of the division is not zero.
Consequently, 2 is not regarded as a factor of 3.
As mentioned earlier, each number has positive and negative factors, and any number’s negative factors are the additive inverse of its positive factors.
Hence, the lists of the positive and negative factors of 3 are given as follows:
Positive Factors of 3 = 1, 3
Negative Factors of 3 = -1, -3
Following the previously used technique, we will put light on the other most commonly celebrated approach, i.e., multiplication, to calculate the factors of 3.
Factors can also be found using the multiplication technique as below:
1 x 3 = 3
3 x 1 = 3
As was previously demonstrated, the numbers 1 and 3 are the well-recognized factors for 3.
Factors of 3 by Prime Factorization
Prime factorization is the technique that focuses on dividing an integer evenly into its prime factors until the result is 1.
The upside-down division methodology is used as the primary approach in the prime factorization technique’s evaluation of the prime factors of a given number, where the division goes on till the final quotient is 1.
The prime factorization of the number 3 is shown below:
3 $\div$ 3 = 1
In addition to the mathematical representation, the visual portrayal of the prime factorization of the number 3 is given as:
Figure 2 – Prime Factorization of 3
At last, the prime factorization of 3 can be written as follows:
Prime Factorization of 3 = 3 x 1
Factor Tree of 3
The pictorial representation of a number’s prime factors is a factor tree. It is called a factor tree because it resembles a tree with many branches joined at the base.
The following steps are to be adopted to construct the factor tree of any given number:
- Place the given number at the top.
- Construct the branches of a tree.
- Mention the prime factors of the given number on each branch.
- Terminate the process by placing the prime factors of the smallest possible number that can have prime factors.
While considering the steps mentioned above, you can construct the factor tree of the number 3 as it is shown below:
Figure 3 – Factor Tree of 3
To ascertain a number’s type, a factor tree is employed. It may show that if an integer is prime, square, or cubic. The factor tree may also be used to compute the LCD and GCD.
It is visible from the factor tree that the number 3 is prime.
Factors of 3 in Pairs
A factor pair is a set of numbers that, when multiplied together, provide the same result as the product of which they are a factor. The pair of factors can be both a set of negative or positive integers.
Luckily, in the case of the number 3, only two factors predict that there will be only a 1-factor pair for 3.
Since the pair of factors is a combination of two factors of a number. In our case, as the number is a highly prime number having only two factors, therefore, the factors of 3 in pairs are given as below:
Figure 4 – Factor pairs of three
In simple words,
Factor Pair of 3 = (1, 3)
As described earlier, the pair of factors are described in terms of positive and negative integers.
Therefore, the negative factor pair of 3 is given as:
Negative Factor Pair of 3 = (-1, -3)
Factors of 3 Solved Examples
This section will provide a few examples to understand the article’s key concepts better.
Example 1
Eric wants to find out the GCD of factors 3 and 9. Can you help him in finding the correct answer?
Solution
The following is the list of factors of 3:
Factors of 3 = 1, 3
The following is the list of factors of 9:
Factors of 9 = 1, 3, 9
The only two common factors amongst the factors of 3 and 9 are 1 and 3. Also, the GCD of factors of 3 and 9 is 3.
Example 2
Emily wants to calculate the sum of all 3 and 27 factors separately. Help her in finding the correct option.
Solution
Let’s first list these factors for calculating the sum of all 3 and 27.
Factors of 3 = 1, 3
Factors of 27 = 1, 3, 9, 27
The sum of these factors is given below:
Sum of Factors of 3 = S1 = 1 + 3
S1 = 4
Sum of Factors of 27 = S2 = 1+3+9+27 = 40
S2 = 40
Hence, the sum of all the factors of the odd numbers 3 and 27 is an even figure of 4 and 40.
Example 3
For a mathematics homework task, Tom is asked to determine the average of all the factors of the number 3. Can you help him find the correct answer and submit the homework on time?
Solution
For calculating the average of all the factors of 3, let’s first list down these factors.
Factors of 3 = 1, 3
Such that,
The average of the set of factors of 3 is achieved by calculating the sum of the factors mentioned above, divided by the total number of factors proposed in the list.
Average of factors = $\dfrac{\text{Sum of factors}}{\text{Total number of factors}}$
Average of factors = $\dfrac{1+3}{2}$
Average = 2
Hence, the average of the factors of 3 is 2.
Images/mathematical drawings are created with GeoGebra.