What are the factor pairs and prime factors of 144?
The concept of factors of 144 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing the factors of 144 helps in solving questions about divisibility, prime factorization, LCM, HCF, and more. This is especially helpful for students in their daily homework, competitive exams, and even day-to-day calculations.
What Are Factors of 144?
A factor of 144 is any whole number that divides 144 exactly without leaving any remainder. In other words, if you multiply two whole numbers and get 144 as the answer, both numbers are factors of 144. This concept is often used in topics such as LCM and HCF, divisibility rules, and square numbers.
Complete List: Factors of 144
Here is the full list of factors of 144, arranged from smallest to largest. You can check that dividing 144 by any of these numbers gives a whole number quotient.
| Factor | Pair |
|---|---|
| 1 | 144 |
| 2 | 72 |
| 3 | 48 |
| 4 | 36 |
| 6 | 24 |
| 8 | 18 |
| 9 | 16 |
| 12 | 12 |
So the positive factors of 144 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.
Prime Factorization of 144 (Step-by-Step)
Prime factorization of 144 means writing it as a product of its prime factors.
1. Divide 144 by 2 (smallest prime): 144 ÷ 2 = 722. Divide 72 by 2: 72 ÷ 2 = 36
3. Divide 36 by 2: 36 ÷ 2 = 18
4. Divide 18 by 2: 18 ÷ 2 = 9
5. Can't divide 9 by 2; try 3: 9 ÷ 3 = 3
6. Divide 3 by 3: 3 ÷ 3 = 1
So, 144 = 2 × 2 × 2 × 2 × 3 × 3 = 24 × 32
The prime factors of 144 are 2 and 3.
Factor Pairs of 144
Factor pairs are sets of two whole numbers that multiply to give 144.
| Factor 1 | Factor 2 |
|---|---|
| 1 | 144 |
| 2 | 72 |
| 3 | 48 |
| 4 | 36 |
| 6 | 24 |
| 8 | 18 |
| 9 | 16 |
| 12 | 12 |
So there are 8 unique pairs (not counting order) and 15 distinct positive factors in total.
Properties and Special Facts About 144
- 144 is a perfect square (since 12 × 12 = 144).
- It is a composite number because it has more than two factors.
- Number of positive factors: 15.
- Even and odd factors both exist (e.g., 2 and 3).
- Sum of all positive factors: 403.
- Prime factorization: 24 × 32.
- The product of all factors is 1447.5 (special property for perfect squares).
Speed Trick: Quickest Way to List All Factors
To quickly find all factors of 144, check divisibility from 1 up to 12 (its square root). For every divisor, write down both that number and 144 divided by it.
- List numbers 1 to 12.
- If 144 ÷ N gives a whole number, N and (144 ÷ N) are both factors.
Example: 144 ÷ 8 = 18 ⇒ So, 8 and 18 are both factors.
Such techniques save time in exams and are actively taught in Vedantu Maths live classes for competitive preparation.
Solved Example: Prime Factorization of 144
1. Start with 1442. Divide by 2 repeatedly: 144 → 72 → 36 → 18 → 9
3. Divide by 3 repeatedly: 9 → 3 → 1
Final Answer: 144 = 24 × 32
Try These Yourself
- Write all odd factors of 144.
- Is 18 a factor of 144?
- List all factor pairs where both numbers are less than 20.
- Find the greatest common factor (GCF) of 144 and 72.
Frequent Errors and Misunderstandings
- Missing repeated factors (e.g., counting 12 × 12 only once).
- Forgetting that both even and odd numbers can be factors.
- Confusing prime factors with all factors.
Relation to Other Maths Concepts
Mastering factors of 144 helps you understand LCM, HCF, divisibility rules, and perfect squares. For example, the LCM and HCF of numbers are found using factors, and knowing 144 is a perfect square will help in square root calculations.
Classroom Tip
Remember: for any perfect square like 144, the square root (here, 12) will always be one of its factor pairs. Vedantu teachers suggest using a factor pair table for faster revision before exams.
We explored factors of 144 — definition, list, fast tricks, solved examples, and more. Continue practicing these concepts with Vedantu for deeper understanding and better scores.
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FAQs on Factors of 144 Explained with Easy Steps
1. What are the factors of 144?
The factors of 144 are the numbers that divide 144 without leaving a remainder. These are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144.
2. How many factors does 144 have?
144 has a total of 15 factors.
3. What are the prime factors of 144?
The prime factorization of 144 is 24 × 32. Therefore, the prime factors are 2 and 3.
4. What is the prime factorization of 144 using exponents?
The prime factorization of 144 using exponents is expressed as 24 x 32.
5. What are the factor pairs of 144?
The factor pairs of 144 are pairs of numbers that multiply to 144. These are: (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), (12, 12).
6. Is 144 a perfect square? If so, what is its square root?
Yes, 144 is a perfect square. Its square root is 12 (since 12 x 12 = 144).
7. How do I find the factors of 144 using the division method?
Divide 144 successively by prime numbers starting from 2, until you reach 1. The prime numbers used are the prime factors. The process would look like this: * 144 ÷ 2 = 72 * 72 ÷ 2 = 36 * 36 ÷ 2 = 18 * 18 ÷ 2 = 9 * 9 ÷ 3 = 3 * 3 ÷ 3 = 1
8. What are some applications of finding the factors of 144 (e.g., in LCM or HCF problems)?
Knowing the factors of 144 is crucial for finding the Least Common Multiple (LCM) and Highest Common Factor (HCF) of 144 with other numbers. You can compare the factors to identify common factors (for HCF) or find the smallest number divisible by both (for LCM).
9. Are all the factors of 144 even numbers?
No. While many factors of 144 are even, some are odd. For example, 1, 3, and 9 are odd factors of 144.
10. How can I use a factor tree to find the prime factorization of 144?
A factor tree visually represents the prime factorization. Start with 144, and break it down into smaller factors until you only have prime numbers. For 144, a possible factor tree would be: 144 → 12 x 12 → 4 x 3 x 4 x 3 → 2 x 2 x 3 x 2 x 2 x 3. This shows the prime factorization as 2 x 2 x 2 x 2 x 3 x 3 or 24 x 32.
11. What are the negative factors of 144?
The negative factors of 144 are simply the negative versions of its positive factors: -1, -2, -3, -4, -6, -8, -9, -12, -16, -18, -24, -36, -48, -72, -144.
12. Explain how the factors of 144 help in solving problems related to divisibility.
Understanding the factors of 144 helps determine if a number is divisible by 144. A number is divisible by 144 only if it contains all the prime factors of 144 (2 and 3) in at least the same quantities (four 2's and two 3's).
