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Factors of 78: Prime Factorization, Methods, Tree, and Examples
The factors of 78 are the natural numbers upon which the natural whole number 78 is completely divisible. When the number 78 is divided under these numbers, the result is a zero remainder and a full number quotient.
Figure 1 – All possible Factors of 78
What Are the Factors of 78?
The factors of 78 are the following: 1, 2, 3, 6, 13, 26, 39, and 78. This set of natural numbers produces zero as a remainder when they act as the dividend for the number 78.In conclusion, the number 78 has a total of 8 factors. These eight factors can pair up to generate 4-factor pairs.How To Calculate the Factors of 78?
You can calculate the factors of 78 from the fundamental method known as the division method. The division method is the most reliable as it produces accurate results and provides a complete set of factors for a number.Division
As the number 78 is a composite number, numerous possibilities exist for various numbers to be a factor of 78. So to narrow down our search, it is beneficial to determine the range in which these factors lie.The range in which factors lie is between the smallest factor 1 and the number which is half of the original number. The number 78 is an even number, so its half is 39. This indicates that to determine the factors of 78, we should look for numbers between 1 and 39. Now let us discuss the application of the division method. The only requirement for this method is that the divisor must yield two things: zero as the remainder and a whole number quotient.If these two results are obtained, the divisor qualifies as a factor. Here is the division showing the various factors of 78:\[ \frac{78}{1} = 78 \]\[ \frac{78}{2} = 39 \]\[ \frac{78}{3} = 26 \]\[ \frac{78}{6} = 13 \]\[ \frac{78}{13} = 6 \]\[ \frac{78}{26} = 3\]\[ \frac{78}{39} = 2 \]\[ \frac{78}{78} = 1\]Hence, the entire list of the factors of 78 is given below:Factors List of 78: 1, 2, 3, 6, 13, 26, 39, and 78There is no restriction that the factors of a number are only required to be positive. These factors can be negative as well. These negative factors are the same as positive ones but have a negative (-) sign next to them.The list of negative factors of 78 is given below:Negative Factors of 78: -1, -2, -3, -6, -13, -26, -39, and -78Factors of 78 by Prime Factorization
The Prime Factorization method is used to evaluate the prime factors of a given number. This method is a division-based method with the condition that only prime numbers can occupy the place of the divisors.The prime numbers carry on the number division, and the division process keeps repeating until the number 1 is achieved at the end. The prime factorization of the number 78 is shown below:78 $\div$ 2 = 39
39 $\div$ 3 = 13
13 $\div$ 13 = 1
The prime factorization of 78 is given below:Prime Factorization of 78 = 2 x 3 x 13
The prime factorization is also represented in the figure given below:
Figure 2 – Prime Factorization of 78
Factor Tree of 78
A Factor Tree is the visual diagram of the prime factorization method. This phenomenon gained its name from the fact that the structure of this diagram resembles a tree. Prime factors are determined through the factor tree.In the factor tree, the number itself acts as the root number. This number splits into two branches, one holding the prime factor and the other having the whole number quotient.This quotient then takes the place of the root number, and the whole process keeps repeating until only prime numbers remain in the end.The factor tree for the number 78 is given below in figure 2:
Figure 3 – Factor Tree of 78
Factors of 78 in Pairs
Factor Pairs are defined as the pair of numbers that multiply with each other, and as the result of this multiplication, they produce the original number. Two numbers can exist within a factor pair.The number 78 has eight factors, which can be divided into 4-factor pairs. Another condition for factor pairs is that both the numbers within the pair must share the same sign.For determining the factor pairs of 78, let’s first list down the factors of 78:Factors of 78 = 1, 2, 3, 6, 13, 26, 39, and 78Evaluating these factor pairs:1 x 78 = 78
2 x 39 = 78
3 x 26 = 78
6 x 13 = 78
So, the factor pairs of 78 are:Factor Pairs of 78: (1, 78), (2, 39), (3, 26), and (6, 13)These are the positive factor pairs. Another possibility for factor pairs is negative factor pairs. The numbers within the pair will have a negative (-) sign for negative factor pairs. So, the negative factor pairs are:Negative Factor Pairs:-1 x -78 = 78
-2 x -39 = 78
-3 x -26 = 78
-6 x -13 = 78
Negative Factor Pairs of 78: (-1, -78), (-2, -39), (-3, -26), and (-6, -13)Factors of 78 Solved Examples
Now that we have discussed everything about the factors of 78, let’s move on to some of the solved examples of factors of 78, which involve factors of 78.Example 1
List the factors of 78 and determine the average of these factors.Solution
In the first part of this example, let’s first list down these factors of 78. These factors are given below:Factors of 78 are the following: 1, 2, 3, 6, 13, 26, 39, and 78Now, let’s consider the second part of this example, which is to determine the average of the factors of 78. The formula for average is given below:\[ Average = \frac{\text{Sum of factors of 78}}{\text{Total number of factors of 78}} \]So for this purpose, let’s calculate the sum of the factors of 78:Sum of factors of 78 = 1 + 2 + 3 + 6 + 13 + 26 + 39 + 78
Sum of factors of 78 = 168
Now we have determined the sum, let’s move on to the average of the factors of 78.Average = 168 / 8
Average = 21
So the average of the factors of 78 is 21.Example 2
Calculate the sum of the odd and even factors of 78 and determine the difference between the two quantities.Solution
For solving this example, let’s first list down the factors of 78:Factors of 78 = 1, 2, 3, 6, 13, 26, 39, and 78Let’s first determine the sum of the odd factors. The odd factors of 78 are given below:Odd Factors of 78 = 1, 3, 13, 39Calculating the sum of these odd factors of 78:Sum of Odd Factors of 78 = 1 + 3 + 13 + 39
Sum of Odd Factors of 78 = 56
Now, let’s move on to the even factors of 78. The even factors of 78 are given below:Even Factors of 78 = 2, 6, 26, 78Calculating the sum of these even factors of 78:Sum of Even Factors of 78 = 2 + 6 + 26 + 78
Sum of Even Factors of 78 = 112
Now that we have calculated the sum of the even and odd factors of 78 let’s move on to calculating the difference between these two quantities.Difference = Sum of even factors – Sum of odd factors
Difference = 112 – 56
Difference = 56
So the difference between the two quantities is 56, which concludes the solution of this example.All images/mathematical drawings are created with GeoGebra.