Factors of 78 Using Prime Factorization
What are the Factors of 78?
If the number 78 is divided by a natural number such that the remainder is zero, then this natural number is known as a factor of 78. Here, the quotient obtained on division is also a factor of 78.
The factors of 78 are 1, 2, 3, 6, 13, 26, 39 and 78 because all these numbers divide the number 78 evenly.
List of the Factors of 78
Divisibility rules and division facts can be used to determine the factors of 78.
Divisor | Is the number a factor of 78? | Multiplication equation |
|---|---|---|
1 | Yes, 1 is a factor of every number. | 1 x 78 = 78 |
2 | Yes, 78 is even. | 2 x 39 = 78 |
3 | Yes, 7 + 8 = 15 is divisible by 3. | 3 x 26 = 78 |
4 | No, 78 \(\div\) 4 = 19 R2 | - |
5 | No, the ones digit is not 0 or 5. | - |
6 | Yes, 78 is even and divisible by 3. | 6 x 13 = 78 |
7 | No, 78 \(\div\) 7 = 11 R1 | - |
8 | No, 78 \(\div\) 8 = 9 R6 | - |
9 | No, 78 \(\div\) 9 = 8 R6 | - |
If we continue the same way, we will notice that the number will repeat on reaching 13. Hence, we can stop checking the factor pairs after 12 as they begin to repeat.
So, the factors of 78 are 1, 2, 3, 6, 13, 26, 39, and 78.
[Note: If we divide 78 by 8, we get the quotient and remainder as 9 and 6, respectively. So, 8 is not a factor of 78. Similarly, you can check for other numbers as well.]
The Prime Factorization of 78
If a natural number can be expressed as a product of prime numbers, these prime numbers are known as the prime factors of that natural number. The process of writing this multiplication of prime factors is known as prime factorization.
The prime factorization of 78 can be represented by using a factor tree as shown below.
So, the prime factorization of 78 is 78 = 2 x 3 x 13, and the prime factors of 78 are 2, 3, and 13.
Factor pairs of 78
A factor pair of a number is a set of any two factors such that their product is the number itself. A factor pair can be negative or positive.
[Note: Product of two negative numbers is positive.]
The factor pairs of 78 are (1, 78), (2, 39), (3, 26), (6, 13).
1 x 78 = 78
2 x 39 = 78
3 x 26 = 78
6 x 13 = 78
Read More:
Positive Factor pairs of 78
Positive Factors of 78 | Positive Factor Pairs of 78 |
|---|---|
1 x 78 | (1, 78) |
2 x 39 | (2, 39) |
3 x 26 | (3, 26) |
6 x 13 | (6, 13) |
Solved Factors of 78 Examples
Example 1: Find the common factors of 68 and 78.
Solution:
Factors of 68 = 1, 2, 4, 17, 34, and 68
Factors of 78 = 1, 2, 3, 6, 13, 26, 39, and 78.
Therefore, the common factors of 68 and 78 are 1 and 2.
Example 2: Find the greatest common factor of 72 and 78.
Solution:
Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72
Factors of 78 = 1, 2, 3, 6, 13, 26, 39, and 78.
So, the common factors of 72 and 78 are 1, 2, 3, and 6.
Therefore, the greatest common factor of 72 and 78 is 6.
Example 3: Tom was studying geometrical shapes and found a triangle of area 39 square centimeters. What are the possible dimensions of the base and height of the triangle? Answer in integers.
Solution :
The area of triangle = \(\frac{1}{2} \times b \times h\) where \(b\) is base and \(h\) is height.
39 = \(\frac{1}{2} \times b \times h\) Given area of the triangle is 39 sq.cm.
78 = \(b \times h\) Multiple both sides by 2
The product of the base and height is 78.
The possible dimensions of base and height are:
Base in centimeters | Height in centimeters |
|---|---|
1 | 78 |
2 | 39 |
3 | 26 |
6 | 13 |
13 | 6 |
26 | 3 |
39 | 2 |
78 | 1 |
Composite numbers are those numbers that have a factor other than 1 and themselves, that is, a composite number has more than two factors.
The division of any positive number by 1 results in zero as the remainder, and the quotient is the number itself. Hence, 1 is a factor of all positive numbers.
Factors of 78 = 1, 2, 3, 6, 13, 26, 39, and 78.
Sum of factors of 78 = 1 +2 + 3 + 6 + 13 + 26 + 39 + 78
= 168