What Are the Factors and Prime Factorization of 49?
The concept of factors of 49 is important in mathematics and is commonly asked in exams, homework, and even Olympiad-level number system problems. Understanding the factors of 49 also helps in learning about divisibility, prime factorization, and perfect squares.
What Is Factors of 49?
A factor of 49 is a whole number that divides 49 exactly, without leaving any remainder. In other words, when 49 is divided by a factor, the result is an integer and the remainder is zero. This concept applies broadly in topics like HCF, LCM, prime factorization, and number properties in competitive exams.
List and Table: All Factors of 49
Let’s find all the numbers that can perfectly divide 49. These numbers are called its factors.
- The factors of 49 are: 1, 7, and 49
This means 49 is divisible by 1, by itself (49), and by 7.
| Factor | Division | Quotient | Remainder |
|---|---|---|---|
| 1 | 49 ÷ 1 | 49 | 0 |
| 7 | 49 ÷ 7 | 7 | 0 |
| 49 | 49 ÷ 49 | 1 | 0 |
Pair Factors of 49
Pair factors or factor pairs are two numbers which multiply together to give 49. Here are the factor pairs:
| Pair | Multiplication |
|---|---|
| (1, 49) | 1 × 49 = 49 |
| (7, 7) | 7 × 7 = 49 |
Notice that because 49 is a perfect square, it has a repeated pair: (7, 7).
Prime Factorization of 49
Prime factorization means writing 49 as a product of prime numbers only. Let’s break it down step by step:
2. 7 itself is a prime number.
3. So, 49 = 7 × 7.
Therefore, the prime factorization of 49 is 7 × 7 (or \( 7^2 \)).
Is 49 Prime or Composite?
49 is not a prime number because it has more than two factors (1, 7, and 49). It is a composite number because it can be divided by numbers other than 1 and itself.
Divisibility Check for 49
To check if a number is a factor of 49, simply divide 49 by that number. If the remainder is zero, it is a factor. Example:
49 ÷ 7 = 7 (remainder 0)
49 ÷ 49 = 1 (remainder 0)
49 ÷ 14 = 3.5 (not a whole number, so 14 is not a factor)
Step-by-Step Method to Find Factors of 49
2. Next, try 2: 49 ÷ 2 = 24.5 (not an integer)
3. Next, try 3: 49 ÷ 3 = 16.33… (not an integer)
4. Try 4: 49 ÷ 4 = 12.25 (not an integer)
5. Try 5 & 6 similarly (not factors)
6. Try 7: 49 ÷ 7 = 7 (integer; so 7 is a factor)
7. Numbers beyond 7 result in repeats (already listed).
So, the complete factors are 1, 7, and 49.
Why Are Factors of 49 Useful?
Factors of 49 help in finding HCF (Highest Common Factor) when paired with another number, LCM (Lowest Common Multiple), and are used in problems involving divisibility and simplifying fractions. Since 49 is a perfect square, these factors are also helpful in geometry and algebra.
For more details on how to find HCF and LCM, see our lesson on HCF of Two Numbers.
Common Student Mistakes
- Thinking 49 is a prime number (it’s composite!)
- Missing the repeated pair (7,7) since 7 × 7 = 49
- Assuming all double-digit numbers less than 49 are factors
- Confusing factors with multiples of 49 (multiples are 49, 98, 147, ...)
Examples and Practice Problems
Example 1: Find the sum of all the factors of 49.
2. Add them: 1 + 7 + 49 = 57
Final Answer: The sum is 57.
Example 2: Is 14 a factor of 49?
2. Since 3.5 is not a whole number, 14 is not a factor.
Example 3: Find the prime factorization of 49.
2. 7 ÷ 7 = 1
So, 49 = 7 × 7
Example 4: Find the HCF of 49 and 84.
2. Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
3. Common factors: 1, 7
4. HCF is 7
Try These Yourself
- Name all factor pairs of 49.
- List all multiples of 49 up to 200.
- Is 21 a factor of 49?
- Find the LCM of 49 and 7.
- Is 49 a perfect square? Why?
Tips and Tricks: Quick Factorization
Since 49 is a perfect square, always check for a repeating factor (here, 7 × 7). For any square number, its square root is a factor, appearing as a repeated pair. Also, remember every number is divisible by 1 and itself.
Related Concepts and Further Learning
Understanding the factors of 49 will help you in related topics like Prime Factorization, Factors of 36 (another square number), and Factors and Multiples.
Classroom Tip
A quick way to remember the factors of a perfect square like 49 is that its square root will always repeat as a pair! Vedantu’s teachers show these patterns visually and through quizzes during live classes.
We explored factors of 49—from the definition, formula, examples, mistakes, and connections to other maths concepts. For step-by-step learning, practice more with Vedantu’s online maths sessions and worksheets.
FAQs on Factors of 49: Definition, Factor Pairs & Prime Factorization
1. What are the factors of 49?
The factors of 49 are the whole numbers that divide 49 exactly without leaving a remainder. These are 1, 7, and 49.
2. What are the factor pairs of 49?
Factor pairs are sets of two numbers that, when multiplied, equal the original number. The factor pairs of 49 are (1, 49) and (7, 7).
3. What is the prime factorization of 49?
Prime factorization expresses a number as a product of its prime factors. The prime factorization of 49 is 7 x 7 or 72.
4. Is 49 a prime number or a composite number?
49 is a composite number because it has more than two factors (1, 7, and 49). A prime number has only two factors: 1 and itself.
5. How do I find the factors of 49 using the division method?
Divide 49 by each whole number starting from 1. If the division results in a whole number with no remainder, the divisor is a factor. For example: 49 ÷ 1 = 49; 49 ÷ 7 = 7; 49 ÷ 49 = 1. Therefore, 1, 7, and 49 are the factors.
6. What numbers divide 49 evenly?
Only 1, 7, and 49 divide 49 evenly (without leaving a remainder).
7. Is 14 a factor of 49?
No, 14 is not a factor of 49 because 49 ÷ 14 = 3.5 (a decimal, not a whole number).
8. How can I use factors of 49 to find the Highest Common Factor (HCF)?
To find the HCF of 49 and another number, list the factors of both numbers and identify the largest factor common to both. For example, to find the HCF of 49 and 84: Factors of 49 are 1, 7, 49; factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. The HCF is 7.
9. How can factors help in solving word problems?
Understanding factors is crucial for solving many word problems involving division, sharing, grouping, and finding common multiples or divisors. For example, if you need to divide 49 sweets equally among a number of children, knowing the factors of 49 helps you determine possible group sizes.
10. What is the sum of all the factors of 49?
The sum of all factors of 49 (1 + 7 + 49) is 57.
11. What is the difference between factors and multiples of 49?
Factors are numbers that divide 49 exactly (1, 7, 49). Multiples are numbers that result from multiplying 49 by a whole number (49, 98, 147, etc.).
12. How is knowing the factors of 49 useful in finding the Lowest Common Multiple (LCM)?
Knowing the prime factorization of a number (like 49 = 7 x 7) is essential when finding the LCM of two or more numbers. The LCM involves identifying the highest powers of all prime factors present in the numbers involved.
