Factors of 72
Factors are numbers that divide a given number exactly without leaving a remainder. Understanding factors is crucial in mathematics, particularly in arithmetic, algebra, and number theory. This article will explore the factors of 72, methods to determine them, their properties, and real-world
applications.
What are the Factors of 72?
A factor of a number is any number that divides it completely without leaving a remainder.
Example:
- 6 is a factor of 72 because 72 ÷ 6 = 12 (no remainder).
- 5 is not a factor of 72 because 72 ÷ 5 = 14.4 (remainder exists).
How to Find the Factors of 72?
To identify the factors of 72, we list all numbers that divide 72 exactly.
Step-by-Step Process:
- Start with 1 and 72: Every number is divisible by 1 and itself.
- Find other divisors: Check numbers that divide 72 without remainder.
- 2 is a factor because 72 ÷ 2 = 36.
- 3 is a factor because 72 ÷ 3 = 24.
- 4 is a factor because 72 ÷ 4 = 18.
- 6 is a factor because 72 ÷ 6 = 12.
- 8 is a factor because 72 ÷ 8 = 9.
Prime Factors of 72
The prime factors of 72 are the factors of 72 that are prime numbers. Unlike the complete list of factors—which includes both prime and composite numbers—prime factors are limited to numbers that are only divisible by 1 and themselves.
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As mentioned earlier, the factors of 72 include 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. However, not all of these are prime. To identify the prime factors of 72, we will use the prime factorization method, as explained in the next section.
Prime Factorization of 72
Prime factorization involves breaking a number down into its smallest prime factors.
Steps for Prime Factorization:
- Start with 72 and divide by the smallest prime number (2).
- 72÷2=36
- Divide 36 by 2 again.
- 36÷2=18
- Divide 18 by 2 again.
- 18÷2=9
- Now, divide 9 by the next smallest prime (3).
- 9÷3=3
- Divide 3 by 3.
- 3÷3=1
Prime Factorization of 72: 72 =2×2×2×3×3 = 23×32
Factor Pairs of 72
Factors can be grouped into pairs whose product is 72.
| Factor 1 | Factor 2 | Product |
| 1 | 72 | 1 × 72 = 72 |
| 2 | 36 | 2 × 36 = 72 |
| 3 | 24 | 3 × 24 = 72 |
| 4 | 18 | 4 × 18 = 72 |
| 6 | 12 | 6 × 12 = 72 |
| 8 | 9 | 8 × 9 = 72 |
So, the factor pairs of 72 are: (1,72), (2,36), (3,24), (4,18), (6,12), and (8,9).
Odd and Even Factors of 72
- Even factors are those divisible by 2: 2, 4, 6, 8, 12, 18, 24, 36, 72
- Odd factors are those not divisible by 2: 1, 3, 9
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Common Factors of 72 with Another Number
To determine common factors, we compare the factors of 72 with another number.
Example: Common Factors of 72 and 54
- Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
- Common factors = 1, 2, 3, 6, 9, 18
Highest Common Factor (HCF) of 72 and Another Number
The HCF is the largest factor that two numbers have in common.
Example: HCF of 72 and 54
Common factors = 1, 2, 3, 6, 9, 18
HCF = 18
Applications of Factors in Real Life
Factors are used in various real-world situations, including:
- Dividing objects equally: When sharing items among groups, factors help determine equal distribution.
- Simplifying fractions: Factors assist in reducing fractions to their simplest form.
- Calculating dimensions: In construction and design, factors help determine possible measurements of objects.
- Finding multiples in schedules: Factors help in identifying common meeting times or event intervals.
Factors of 72 FAQs
What are the factors of 72?
The factors of 72 are- 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
What are the prime factors of 72?
The prime factors of 72 are 2 and 3. When written as prime factorization, 72 = 2 x 2 x 2 x 3 x 3 = 23 x 32
What is the sum of the factors of 72?
Adding all the factors of 72: 1+2+3+4+6+8+9+12+18+24+36+72=195. So, the sum of the factors is 195.
What is the greatest common factor (GCF) of 72 and 60?
- Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Common factors = 1, 2, 3, 4, 6, 12
- Greatest common factor (GCF) = 12
How many factors does 72 have?
There are 12 factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
Are all factors of 72 divisible by 2?
No, only even factors are divisible by 2. The odd factors of 72 are 1, 3, and 9, which are not divisible by 2.