What are the Factor Pairs and Prime Factors of 105?
The concept of factors of 105 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding factors and their properties is essential for solving questions on divisibility, multiples, HCF, LCM, and even for word problems that appear in competitive exams and school tests. Let’s explore everything you need to know about the factors of 105 with easy visuals, stepwise explanations, and exam-friendly tricks.
What Is Factors of 105?
The factors of 105 are all numbers that divide 105 exactly without leaving any remainder. In other words, if you multiply any two of these numbers, you get 105. You’ll find this concept applied in areas such as HCF/LCM, prime factorization, and many daily scenarios involving equal grouping or splitting objects.
How to Find the Factors of 105?
To find the factors of 105, check which numbers divide 105 with no remainder. Here are the steps:
- Start with 1: 105 ÷ 1 = 105, so 1 and 105 are factors.
- Check divisibility by 2: 105 is odd, so skip.
- Check 3: 105 ÷ 3 = 35, remainder 0 → 3 and 35 are factors.
- Check 5: 105 ÷ 5 = 21, remainder 0 → 5 and 21 are factors.
- Check 7: 105 ÷ 7 = 15, remainder 0 → 7 and 15 are factors.
- Continue till √105 (~10.25). All possible unique factors are done.
So, the complete list of factors of 105 is: 1, 3, 5, 7, 15, 21, 35, 105.
All Factors of 105 - Table
| Factor | Is Prime? | Pair |
|---|---|---|
| 1 | No | (1, 105) |
| 3 | Yes | (3, 35) |
| 5 | Yes | (5, 21) |
| 7 | Yes | (7, 15) |
| 15 | No | (15, 7) |
| 21 | No | (21, 5) |
| 35 | No | (35, 3) |
| 105 | No | (105, 1) |
Factors of 105 in Pairs
Factor pairs are two numbers that multiply together to give 105. Let’s list them:
| Pair | Product |
|---|---|
| (1, 105) | 105 |
| (3, 35) | 105 |
| (5, 21) | 105 |
| (7, 15) | 105 |
Negative pairs (multiplying negatives gives positive): (-1, -105), (-3, -35), (-5, -21), (-7, -15)
Prime Factorization of 105
The prime factors of 105 break 105 down to its basic building blocks using only prime numbers. Let’s use a factor tree:
1. 105 ÷ 3 = 35.2. 35 ÷ 5 = 7.
3. 7 is already prime.
So, 105 = 3 × 5 × 7
All three—3, 5, and 7—are prime numbers. This is helpful for problems on LCM, HCF, and more. For a visual factor tree, start with 105 and break it by dividing by the smallest prime till only primes are left at the branches.
Properties and Application of Factors of 105
- 105 is a composite number since it has factors other than 1 and itself.
- It is not a prime.
- The sum of all positive factors = 1 + 3 + 5 + 7 + 15 + 21 + 35 + 105 = 192.
- Applications include simplifying fractions, dividing objects equally, finding common denominators, and preparing for exams.
- Prime factorization helps in finding HCF (Highest Common Factor) and LCM (Lowest Common Multiple) with other numbers.
Speed Trick: Check Factors of 105 Quickly!
Mental Math Quick Check:
- If the sum of digits (1+0+5=6) is a multiple of 3, then 3 is a factor.
- The last digit is 5, so 5 is a factor.
- Since 1 × 3 × 5 × 7 = 105 (all primes), you can break it into small, easy steps.
These are useful when you want to save time in competitive exams and quizzes. Vedantu’s live classes provide many more such tricks for other numbers too!
Try These Yourself
- Find all even factors of 105. (Hint: is there any?)
- What are the common factors between 105 and 35?
- If you split 105 mangoes equally into baskets, what is the maximum basket size using only whole numbers?
- Does 21 divide 105 exactly? Show the working.
Frequent Mistakes to Avoid
- Missing out 1 or 105 as a factor.
- Counting only prime factors and forgetting composite ones like 15 or 21.
- Confusing multiples (like 210, 315) with factors (which are smaller than or equal to 105).
Relation to Other Concepts
The idea of factors of 105 connects closely with prime factors, factors of 35, factors of 120, and factors and multiples. Mastering these will help you solve advanced problems in number theory, fractions, and algebra.
Classroom Tip
A quick way to remember the prime factors of 105 is to think of the first three odd primes: 3, 5, and 7—since 3 × 5 × 7 = 105. Vedantu’s teachers often use this trick to help students build mental math skills.
We explored factors of 105—from the definition, pairs, prime factorization, common mistakes, to exam tricks and real applications. Keep practicing with Vedantu and refer to related topics like prime numbers, prime factorization, and factors of 60 to become confident in number-related maths problems!
FAQs on Factors of 105 Explained with Examples
1. What are the factors of 105?
The factors of 105 are the numbers that divide 105 without leaving a remainder. These are: 1, 3, 5, 7, 15, 21, 35, and 105. Their negative counterparts (-1, -3, -5, -7, -15, -21, -35, -105) are also factors.
2. What is the prime factorization of 105?
The prime factorization of 105 expresses it as a product of its prime factors. It is 3 × 5 × 7. This means 3, 5, and 7 are the only prime numbers that multiply to give 105.
3. What are the factor pairs of 105?
Factor pairs of 105 are pairs of numbers that multiply to 105. The positive pairs are: (1, 105), (3, 35), (5, 21), and (7, 15). Negative counterparts also exist (-1, -105), (-3, -35), (-5, -21), (-7, -15).
4. How do you find the factors of 105 using the division method?
To find factors using division, systematically divide 105 by each integer starting from 1. If the division results in a remainder of 0, the divisor is a factor. For example: 105 ÷ 1 = 105; 105 ÷ 3 = 35; 105 ÷ 5 = 21; 105 ÷ 7 = 15; and so on, until you reach 105 itself.
5. How do I use a factor tree to find the prime factors of 105?
A factor tree visually represents the prime factorization. Start with 105. Break it down into any two factors (e.g., 3 and 35). Continue breaking down composite factors until you're left with only prime numbers (numbers divisible only by 1 and themselves). For 105, the tree will eventually show the prime factors: 3, 5, and 7.
6. What is the sum of all the factors of 105?
To find the sum, add all the factors of 105 (both positive and negative): 1 + 3 + 5 + 7 + 15 + 21 + 35 + 105 + (-1) + (-3) + (-5) + (-7) + (-15) + (-21) + (-35) + (-105) = 0. Note that the sum of positive factors is 186 and the sum of negative factors is -186.
7. Is 105 a prime or composite number?
105 is a **composite number** because it has factors other than 1 and itself (e.g., 3, 5, 7).
8. What are some common mistakes students make when finding factors?
Common mistakes include: missing factors, including non-whole numbers as factors, only listing positive factors, and not understanding the difference between factors and multiples.
9. How are factors used in solving real-world problems?
Factors are used in various applications, including dividing items equally, determining the dimensions of rectangles with a given area, understanding ratios and proportions, and simplifying fractions.
10. How can I quickly check if a number is a factor of 105?
Use divisibility rules. Since 105 = 3 × 5 × 7, a number is a factor of 105 if it is a factor of 3, 5, or 7, or a combination thereof. For example, 15 (3 × 5) is a factor, but 4 is not.
11. What is the difference between factors and multiples?
Factors are numbers that divide a given number exactly, while multiples are numbers obtained by multiplying a given number by an integer. For example, the factors of 105 are 1, 3, 5, 7, etc., while the multiples of 105 are 105, 210, 315, etc.
12. How are the factors of 105 relevant to finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM)?
Understanding the prime factorization of 105 (3 x 5 x 7) is crucial for efficiently calculating the HCF and LCM with other numbers. The HCF is found by identifying common prime factors, while the LCM uses all prime factors, considering the highest power of each.
