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 include all whole numbers that divide 144 without any remainder. These are
- 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72,
- and 144
2. How do you find the factors of 144?
To find the factors of 144, divide 144 by whole numbers, beginning with 1 and ending with 144. If the division results in a whole number with no remainder, both the divisor and quotient are factors of 144.
3. Is 144 a perfect square?
Yes, 144 is a perfect square because $12 \times 12 = 144$. Its square root $\sqrt{144}$ equals 12, showing that 144 comes from multiplying a whole number by itself, which makes it a perfect square number.
4. What is the prime factorization of 144?
The prime factorization of 144 breaks the number down into only prime numbers: $144 = 2^4 \times 3^2$. This means 144 is made by multiplying 2 four times and 3 two times together.
5. How many factors does 144 have?
The number 144 has 15 factors. To find this, add one to each exponent in the prime factorization and multiply: (4+1)$\times$(2+1) = 5$\times$3 = 15. So, 144 has a total of fifteen factors.
6. Are all factors of 144 even?
Not all factors of 144 are even, though most are. The odd factors are
- 1, 3, and 9
7. What pairs of numbers multiply to give 144?
Pairs of numbers that multiply to give 144 as factor pairs include
- 1 × 144
- 2 × 72
- 3 × 48
- 4 × 36
- 6 × 24
- 8 × 18
- 9 × 16
- 12 × 12
8. What are the common factors of 144 and 100?
The common factors of 144 and 100 are the numbers that divide both without remainder. These shared factors are
- 1, 2, 4, 8, 16
9. Can 144 be divided by 5?
No, 144 cannot be divided evenly by 5. If you divide 144 by 5, you get 28.8 with a remainder, which means 5 is not a factor of 144, and the division is not exact.
10. Are there negative factors of 144?
Every positive factor of 144 has a negative pair. For example, both 12 and –12 multiply with their corresponding pairs (12 × 12 or (–12) × (–12)) to make 144. So, 144 also has negative factors.
11. What is the sum of all factors of 144?
The total of all factors of 144 is 403. If you add 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144 together, the sum equals 403.
