How to Find the Factors of 105 Step by Step with Examples
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 Complete Guide with Factor Pairs and Prime Factorization
1. What are the factors of 105?
The factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105.
- A factor is a number that divides 105 exactly without leaving a remainder.
- Since 105 is a composite number, it has more than two factors.
- These factors come from its prime factorization: 105 = 3 × 5 × 7.
2. How do you find the factors of 105?
You can find the factors of 105 by using prime factorization or division method.
- Step 1: Divide 105 by small prime numbers: 105 ÷ 3 = 35.
- Step 2: Factor 35 as 5 × 7.
- Step 3: So, 105 = 3 × 5 × 7.
- Step 4: Form all combinations: 1, 3, 5, 7, 15, 21, 35, 105.
3. Is 105 a prime or composite number?
The number 105 is a composite number because it has more than two factors.
- A prime number has exactly two factors: 1 and itself.
- 105 has eight factors: 1, 3, 5, 7, 15, 21, 35, 105.
- Therefore, 105 is not prime.
4. What is the prime factorization of 105?
The prime factorization of 105 is 3 × 5 × 7.
- Divide 105 by 3 → 35.
- Divide 35 by 5 → 7.
- 7 is already a prime number.
- So, 105 = 3 × 5 × 7.
5. How many factors does 105 have?
The number 105 has 8 positive factors.
- Prime factorization: 105 = 3¹ × 5¹ × 7¹.
- Use the formula for total factors: (1+1)(1+1)(1+1).
- That equals 2 × 2 × 2 = 8.
6. What are the factor pairs of 105?
The factor pairs of 105 are (1, 105), (3, 35), (5, 21), and (7, 15).
- A factor pair multiplies to give 105.
- Example: 3 × 35 = 105.
- Each smaller factor pairs with a corresponding larger factor.
7. What are the common factors of 105 and 35?
The common factors of 105 and 35 are 1, 5, 7, and 35.
- Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105.
- Factors of 35: 1, 5, 7, 35.
- The shared numbers are the common factors.
8. What is the greatest common factor (GCF) of 105 and 70?
The greatest common factor of 105 and 70 is 35.
- Prime factorization of 105 = 3 × 5 × 7.
- Prime factorization of 70 = 2 × 5 × 7.
- Common prime factors: 5 and 7.
- Multiply them: 5 × 7 = 35.
9. Is 105 divisible by 3, 5, and 7?
Yes, 105 is divisible by 3, 5, and 7 exactly.
- 105 ÷ 3 = 35.
- 105 ÷ 5 = 21.
- 105 ÷ 7 = 15.
- This confirms 3, 5, and 7 are factors of 105.
10. What is the smallest and greatest factor of 105?
The smallest factor of 105 is 1 and the greatest factor is 105.
- Every positive integer has 1 as its smallest factor.
- The number itself is always its greatest factor.
- This rule applies to all natural numbers.
