Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 as they can divide 72 without leaving any remainder. We know that the factors are those numbers that divide the given number without leaving any remainder or that can be multiplied together with other factors to give the result of the number itself.
In this article, we will answer different questions related to factors like “What are Factors?”, “What are some Factors of 72?”, and “How to Find Factors of 72?".
What is a Factor?
A factor is a number that divides the given number without any remainder.
The factors of a number can either be positive or negative which can divide the given number completely without leaving a remainder. For example, both 3 and -3 are factors of 9, since 9 divided by 3 equals 3, and 9 divided by -3 equals -3, with no remainder in either case.
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What are the Factors of 72?
Factors of 72 are those numbers that divide the number without leaving any remainder i.e. 72 ÷ Factor, then remainder = 0.
In other words, we can also say that factors are those numbers that multiply together to give a product equal to 72.
For example: 12 and 6 are factors of 72 because the product of 12 and 6 are 72
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So, the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
| Factors of 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72 |
|---|---|
| Pairs of Factors | (1, 72), (2, 36), (3,24), (4, 18), (6, 12) and (8,9) |
| Prime Factorization of 72 | 72 = 2 × 2 × 2 × 3 × 3 |
Sum of Factors of 72 | 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195 |
Product of Factors | 1 × 2 × 3 × 4 × 6 × 8 × 9 × 12 × 18 × 24 × 36 × 72 = 139314069504 |
Number of Factors of 72 | 12 |
All Factors of 72
List of all factors of 72 include:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
How to Find Factors of 72
To find the factors of 72, to find the factors of 72 you need to find out all the whole numbers that can divide 72 without leaving a remainder.
| Divisor | Is the number a factor of 72? | Pairs |
|---|---|---|
| 1 | we know that 1 is a factor of every number | 1 × 72 = 72 |
| 2 | 72 is divisible by 2 | 2 × 36 = 72 |
| 3 | As 7 + 2 =9, so 72 divisible by 3 | 3 × 24 = 72 |
| 4 | Yes, 72 is divisible by 4 | 4 × 18 = 72 |
| 5 | As ones digit of 72 is not zero or 5. | - |
| 6 | 72 is divisible by 6 | 6 × 12 = 72 |
| 7 | No | - |
| 8 | As 72 is divisible by 8 and remainder=0. | 8 × 9 = 72 |
Prime Factorization of 72
Prime factorization is method of expressing any number as the product of prime numbers.
To find out prime factorization of 72, divide 72 by its prime factors (you can begin by dividing by its smallest factor that is 2), until we get the result as 1.
Step 1: We know that 72 is even, so first we will divide by 2. We will get 36
Step 2: Again 36 is even, so divide by 2 we get 18.
Step 3: 18 is also even so when divided by 2 we get 9.
Step 4: Now, for 9 we will divide it by next prime number that is 3. we get .
Step 5: 3 when divided by 3 give 1. and we will end the prime factorization.
This process of prime factorization can be illustrated as follows:
So, prime factorization of 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32
Prime Factors of 72
The prime factors of a number are the prime numbers that exactly divide the given number without leaving a remainder. In the case of 72, its prime factors are 2 and 3.
Factor Tree of 72
A factor tree is a diagram that breaks down a number into its prime factors. The factor tree of 72 can be illustrated as:
Factor Pairs of 72
Factors pairs are those number when multiplied together gives 72. Those pairs can be negative as well as positive.
Positive Factor Pairs of 72 | Negative Factor Pairs of 72 |
|---|---|
| 1 × 72 | (-1) × (-72) |
| 2 × 36 | (-2) × (-36) |
| 3 × 24 | (-3) × (-24) |
| 4 × 18 | (-4) × (-18) |
| 6 × 12 | (-6) × (-12) |
| 8 × 9 | (-8) × (-9) |
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Solved Examples on Factors of 72
Example 1: What are the factor common between 6 and 72.
Solution:
Factors of 6: 1, 2, 3, 6.
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
So common factors are 1, 2, 3, and 6.
Example 2: What is the sum of all factors of 72.
Solutions:
Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Sum = 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 193.
Example 3: What is the sum of odd factors of 72.
Solution:
Odd factors of 72 are 1, 3, 9.
Sum = 1 + 3 + 9 = 13.
Example 4: If a is a factor of 72 and a <12, what is the largest possible value of a?
Solution:
all the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Ass we can see the largest possible value of a less than 12 is 9.
So, a = 9.
Practice Questions on Factors of 72
Question 1: How many factors does 72 have?
Question 2: Identify the prime factors of 72.
Question 3: Find the sum of all the factors of 72.
Question 4: Determine the product of all the factors of 72.
Question 5: What is the greatest common divisor (GCD) of 72 and 36?
Question 6: What is the least common multiple (LCM) of 72 and 24?
Question 7: If you group the factors of 72 into pairs that multiply to 72, what are the pairs?
Question 8: Are there more even or odd factors of 72? List them.
Conclusion
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72, and knowing these factors is useful in various mathematical tasks such as division, simplifying fractions, and solving problems. Finding the factors of 72 helps us better understand how numbers can be broken down into their smaller components.which is important not only in basic math but also in more advanced topics like algebra.