What Are the Factors and Prime Factorization of 35?
The concept of factors of 35 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how to find factors, their properties, and their use in problem-solving helps students prepare confidently for board exams and competitive tests. Let’s explore everything you need to know about the factors of 35 in a step-by-step, easy-to-understand way.
What Are the Factors of 35?
A factor of a number is any whole number that divides the given number exactly with no remainder. Factors of 35 are the numbers you can multiply in pairs to get 35 or divide 35 by to get another whole number. You’ll find this concept applied in areas such as multiples, divisibility rules, and word problem solving.
List of All Factors of 35
The positive factors of 35 are:
| Factor | Explanation |
|---|---|
| 1 | 1 × 35 = 35 |
| 5 | 5 × 7 = 35 |
| 7 | 7 × 5 = 35 |
| 35 | 35 × 1 = 35 |
So, the factors of 35 are 1, 5, 7, and 35.
Factor Pairs of 35
Factor pairs are two numbers that, when multiplied together, give you 35. Here are all the factor pairs of 35:
| Positive Pair | Negative Pair |
|---|---|
| (1, 35) | (-1, -35) |
| (5, 7) | (-5, -7) |
Prime Factorization of 35
To express 35 as a product of prime numbers, follow the prime factorization method:
| Step | Action |
|---|---|
| 1 | 35 is not divisible by 2 or 3 (they are not factors). |
| 2 | Try 5: 35 ÷ 5 = 7 (no remainder, so 5 is a prime factor). |
| 3 | Now divide 7 by 7: 7 ÷ 7 = 1 (done, since 7 is also prime). |
So, the prime factorization of 35 is 5 × 7.
Quick Division Method for Factors of 35
To find all factors of 35 efficiently, use the division method—divide 35 by each whole number from 1 to 35. If the result is a whole number with no remainder, the divisor is a factor.
2. 35 ÷ 2 = 17.5 (not a factor)
3. 35 ÷ 5 = 7 (factor)
4. 35 ÷ 7 = 5 (factor)
5. 35 ÷ 35 = 1 (factor)
Therefore, only 1, 5, 7, and 35 give whole numbers on division. These are the factors of 35.
Divisibility Properties of 35
- 35 is an odd composite number.
- It is evenly divisible by 1, 5, 7, and itself.
- All factors of 35 are odd numbers.
- Since more than two factors exist, 35 is not a prime number.
Step-by-Step Example Problem
Question: List all common factors of 35 and 25.
2. Factors of 25 = 1, 5, 25
3. The common factors (present in both) are 1 and 5.
Final Answer: The common factors of 35 and 25 are 1 and 5.
Speed Trick to List Factors Quickly
If you know the multiplication tables up to 10, spotting factors becomes faster. Use the fact that factors come in pairs—after you reach the square root of 35 (≈5.9), you’ve found all factors. This helps save time during exams.
Try These Yourself
- What is the sum of all factors of 35?
- Is 6 a factor of 35?
- Find factors of numbers from 30 to 40 and compare them with 35.
- List all even numbers that are also factors of 35 (hint: is there any?).
Frequent Errors and Misunderstandings
- Forgetting that 1 and the number itself are always factors of a whole number.
- Thinking 35 is prime just because 5 and 7 are prime (35 is composite).
- Missing negative factors: -1, -5, -7, -35 are also valid factors in higher-level maths.
Related Maths Topics to Explore
The idea of factors of 35 connects closely with these topics—which you can learn more about at Vedantu:
Classroom Tip
A helpful way to remember factor pairs: Draw a “rainbow” connecting number pairs (like 1–35 and 5–7) above your number line. Vedantu’s teachers use such tricks and visuals to boost memory and reduce silly errors.
We explored factors of 35—what they are, how to calculate them, shortcut methods, common mistakes, and where this knowledge is used in higher topics. With Vedantu’s stepwise explanations and practice examples, you’ll build confidence and accuracy for exams and beyond. Keep practicing factors to master all related maths skills!
FAQs on Factors of 35: List, Prime Factorization & Common Questions
1. What are the factors of 35?
The factors of 35 are the numbers that divide 35 without leaving a remainder. These are 1, 5, 7, and 35. We can also include their negative counterparts: -1, -5, -7, and -35.
2. What is the prime factorization of 35?
The prime factorization of 35 is 5 × 7. This means 5 and 7 are the only prime numbers that multiply to give 35.
3. How many factors does 35 have?
35 has four positive factors (1, 5, 7, 35) and four negative factors (-1, -5, -7, -35). Therefore, it has a total of eight factors.
4. Is 35 a prime or composite number?
35 is a composite number because it has more than two factors.
5. What are the factor pairs of 35?
The positive factor pairs of 35 are (1, 35) and (5, 7). The negative factor pairs are (-1, -35) and (-5, -7).
6. How do I find the factors of 35 using the division method?
Divide 35 by each integer, starting from 1, until you reach 35. If the division results in a whole number (no remainder), that integer is a factor. For example: 35 ÷ 1 = 35, 35 ÷ 5 = 7, 35 ÷ 7 = 5, 35 ÷ 35 = 1.
7. How do I find the factors of 35 using a factor tree?
Start by expressing 35 as a product of two factors (e.g., 5 × 7). Since both 5 and 7 are prime numbers, you've found the prime factorization. The factors are 1, 5, 7, and 35 (and their negative counterparts).
8. What are the common factors of 35 and 25?
The factors of 35 are 1, 5, 7, 35. The factors of 25 are 1, 5, 25. The common factors of 35 and 25 are 1 and 5.
9. What is the greatest common factor (GCF) of 35 and 49?
The factors of 35 are 1, 5, 7, 35. The factors of 49 are 1, 7, 49. The greatest common factor of 35 and 49 is 7.
10. Is 7 a factor of 35?
Yes, 7 is a factor of 35 because 35 divided by 7 equals 5 with no remainder.
11. Explain the difference between factors and multiples of 35.
Factors of 35 are numbers that divide 35 evenly (1, 5, 7, 35). Multiples of 35 are numbers that result from multiplying 35 by an integer (35, 70, 105, etc.).
12. What are some real-life examples where understanding factors of 35 is useful?
Understanding factors is useful in various real-life situations, such as dividing 35 items equally among a group, or determining the dimensions of a rectangle with an area of 35 square units. It is also crucial in higher-level math concepts like finding the greatest common factor (GCF) and least common multiple (LCM).
