What are the Pair Factors and Prime Factorization of 98?
The concept of factors of 98 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how to find factors helps students in topics such as HCF, LCM, divisibility, and problem-solving questions. This topic is especially useful for middle school classes and competitive exams.
What Are the Factors of 98?
A factor of 98 is a whole number that divides 98 exactly without leaving any remainder. In mathematics, these are the numbers you can multiply in pairs to get 98. Concepts like factors of a number, prime numbers, and multiples are all related to factors of 98. You’ll find this idea applied in calculating HCF, LCM, and simplifying fractions.
Key Formula for Factors of 98
Here’s the standard formula:
\( \text{Factors of 98} = \{ n : 98 \div n = 0, \ n \in \mathbb{Z}^+ \} \)
Cross-Disciplinary Usage
Factors of 98 are not only useful in Maths but also play an important role in Physics (like measurements and ratios), Computer Science (like programming divisibility), and daily logical reasoning. Students preparing for JEE, NTSE, or school Olympiads will see their relevance in many word problems and MCQs.
How to Find the Factors of 98?
Finding the factors of 98 is easy if you follow a step-by-step process:
- Start with 1 and 98 (because every number is divisible by 1 and itself).
- Check small numbers up to √98 (about 9.89).
- Divide 98 by these numbers and see where you get whole numbers as answers.
- Write each pair you find as a factor pair.
Step-by-step:
1. 98 ÷ 1 = 98 → Factors: 1 and 982. 98 ÷ 2 = 49 → Factors: 2 and 49
3. 98 ÷ 7 = 14 → Factors: 7 and 14
4. 98 ÷ any other whole number between 1 and 9 (except these) gives a remainder.
All Factors of 98: Table
| Factor | Division Check |
|---|---|
| 1 | 98 ÷ 1 = 98 (whole) |
| 2 | 98 ÷ 2 = 49 (whole) |
| 7 | 98 ÷ 7 = 14 (whole) |
| 14 | 98 ÷ 14 = 7 (whole) |
| 49 | 98 ÷ 49 = 2 (whole) |
| 98 | 98 ÷ 98 = 1 (whole) |
Pair Factors of 98
Pair factors are two numbers that multiply to give 98. These are important for quick checks, MCQs, and reasoning sums.
| Positive Pair | Product |
|---|---|
| (1, 98) | 1 × 98 = 98 |
| (2, 49) | 2 × 49 = 98 |
| (7, 14) | 7 × 14 = 98 |
Negative pairs (for advanced learners): (-1, -98), (-2, -49), (-7, -14), since multiplying two negatives gives a positive.
Prime Factorization of 98
The prime factorization of 98 breaks it into only prime numbers (numbers greater than 1 with only 1 and itself as factors).
1. Start with the smallest prime: 98 is even, so divide by 2.2. 98 ÷ 2 = 49
3. 49 is not divisible by 3 or 5... but by 7.
4. 49 ÷ 7 = 7
5. 7 is prime, so stop here.
So, 98 = 2 × 7 × 7 or 2 × 72.
You can draw a quick factor tree for visual memory:
98
/ \
2 49
/ \
7 7
Speed Tricks for Factors of 98
Quick Tip: Since 98 ends in 8, it’s divisible by 2. Divide by 2 first, then check the result for other factors like 7 and its multiples. Also, 98 = 2 × 49 (and 49 is 7 × 7). Remember, if a number is divisible by both 2 and 7, it’s definitely a factor of 98!
Try These Yourself
- What are all the factors of 98?
- Which of these is NOT a factor of 98: 7, 14, 8, or 49?
- Find the sum of all pair factors of 98.
- Is 21 a factor of 98?
Frequent Errors and Misunderstandings
- Forgetting to check all numbers up to square root of 98.
- Thinking 4 or 8 is a factor (98 ÷ 4 is not a whole number).
- Mixing up multiples and factors.
- Writing non-whole numbers as factors (like 49/2).
Relation to Other Concepts
The idea of factors of 98 connects with prime factorization, HCF and LCM, and composite numbers. When you know the factors, you can solve problems related to fractions, ratios, divisibility, and more advanced maths topics.
Classroom Tip
A quick way to remember the prime factorization of 98 is this: “Try dividing by 2; then, since the result is a square number (49), check for square roots – 7 × 7. Drawing a factor tree helps you spot all these visually. Vedantu’s teachers use this approach in class for rapid learning!
Wrapping It All Up
We explored factors of 98—from definition, key formula, listing, step-by-step solving, to frequent mistakes and links with other maths topics. Keep practicing these steps with help from Vedantu to master factors and ace your exams!
Further Reading & Related Topics
FAQs on Factors of 98 Explained with Examples
1. What are the factors of 98?
The factors of 98 are the numbers that divide 98 without leaving a remainder. These are: 1, 2, 7, 14, 49, and 98.
2. What is the prime factorization of 98?
The prime factorization of 98 expresses it as a product of prime numbers. It is 2 × 7 × 7, or 2 × 72.
3. How many factors does 98 have?
The number 98 has six positive factors: 1, 2, 7, 14, 49, and 98. It also has six negative factors (-1, -2, -7, -14, -49, -98).
4. What are the factor pairs of 98?
Factor pairs are pairs of numbers that multiply to give 98. The positive factor pairs are (1, 98), (2, 49), and (7, 14). Negative pairs also exist (-1,-98), (-2,-49), (-7,-14).
5. Is 4 a factor of 98?
No, 4 is not a factor of 98 because 98 divided by 4 leaves a remainder of 2.
6. How do I find the factors of a larger number like 98?
To find all factors, systematically check for divisibility starting from 1. You only need to check up to the square root of the number (√98 ≈ 9.9). If a number is a factor, its pair (98 divided by that number) will also be a factor.
7. What is the difference between factors and multiples of 98?
Factors are numbers that divide 98 evenly, while multiples are numbers obtained by multiplying 98 by integers. For example, factors of 98 include 1, 2, 7, etc., while multiples include 98, 196, 294, etc.
8. How are factors of 98 useful in solving math problems?
Understanding factors is crucial for simplifying fractions, finding the **highest common factor (HCF)** and **lowest common multiple (LCM)**, and solving problems related to divisibility.
9. Can negative numbers be factors of 98?
Yes. For every positive factor, there's a corresponding negative factor. For instance, if 2 is a factor, then -2 is also a factor because (-2) * (-49) = 98.
10. What are some common mistakes students make when finding factors?
Common mistakes include forgetting to include 1 and the number itself as factors, not checking for negative factors, and incorrectly applying divisibility rules.
11. Is 98 a prime or composite number?
98 is a **composite number** because it has more than two factors (1 and itself).
12. Explain the factor tree method for finding prime factors.
The factor tree method visually breaks down a number into its prime factors. Start with the original number (98). Find a pair of factors, then break down composite factors until you only have prime numbers remaining. For 98, this would look like: 98 = 2 x 49 = 2 x 7 x 7.
