What are the factors of 63 and how to find them using prime factorization
The concept of factors of 63 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Are the Factors of 63?
A factor of 63 is any whole number that divides 63 exactly without leaving any remainder. In other words, if you multiply two numbers and get 63, both of those numbers are factors of 63. Factors appear in maths exams, factor-pair questions, and real-life scenarios such as arranging objects evenly into groups or rows. They also help with topics like common factors, prime factorization, and finding the LCM and HCF of numbers.
List of All Factors of 63
The factors of 63 are the numbers that divide 63 exactly, leaving no remainder. If you list and test all numbers from 1 up to 63, you get the following:
| Number | Can Divide 63? | Result |
|---|---|---|
| 1 | Yes | 63 / 1 = 63 |
| 3 | Yes | 63 / 3 = 21 |
| 7 | Yes | 63 / 7 = 9 |
| 9 | Yes | 63 / 9 = 7 |
| 21 | Yes | 63 / 21 = 3 |
| 63 | Yes | 63 / 63 = 1 |
Hence, the factors of 63 are: 1, 3, 7, 9, 21, and 63. These numbers divide 63 exactly with no remainder.
Factor Pairs of 63
Factor pairs are two numbers that multiply together to give 63. For 63, the factor pairs are:
| Pair | Multiplication |
|---|---|
| (1, 63) | 1 × 63 = 63 |
| (3, 21) | 3 × 21 = 63 |
| (7, 9) | 7 × 9 = 63 |
Negative numbers also form pairs: (-1, -63), (-3, -21), and (-7, -9).
Prime Factorization of 63
Prime factorization means breaking a number down to only its prime number factors. For 63, we do:
1. Start with the smallest prime, 2. (63 ÷ 2 = 31.5, not a whole number)2. Next is 3. (63 ÷ 3 = 21, whole number)
3. Divide 21 by 3 again (21 ÷ 3 = 7, whole number)
4. 7 is a prime number.
So the prime factorization of 63 is 3 × 3 × 7 (or 32 × 7).
Is 63 a Prime or Composite Number?
A prime number has only two factors: 1 and itself. However, 63 has six positive factors (1, 3, 7, 9, 21, 63), so 63 is a composite number. It is not prime because it can be divided by numbers other than 1 and 63.
How to Find Factors of 63 – Stepwise Method
You can use these steps to find all factors quickly during exams:
1. Start from 1 and check each number up to 63.2. If 63 ÷ number = whole number, the number is a factor.
3. Repeat for all numbers up to the square root of 63 (about 7.9) for faster checking.
4. List each factor you find. Write its pair: (number, 63 ÷ number).
Tip: Since 63 ends with an odd digit, skip even numbers. If the sum of digits (6 + 3 = 9) is divisible by 3, then 3 is a factor—quick divisibility trick!
Factor Tree of 63
A factor tree visually breaks down 63 into primes:
↙ ↘
3 21
↙ ↘
3 7
This shows 63 = 3 × 3 × 7. Try drawing such a tree for practice!
Common Factors of 63 with Other Numbers
To find common factors, list the factors of each number and check for overlap:
| Number | Factors | Common with 63 |
|---|---|---|
| 63 | 1, 3, 7, 9, 21, 63 | All itself |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 1, 3, 9 |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 1, 3, 9 |
| 54 | 1, 2, 3, 6, 9, 18, 27, 54 | 1, 3, 9 |
| 27 | 1, 3, 9, 27 | 1, 3, 9 |
Quick Reference Table: Factors of Key Numbers
| Number | All Factors | Prime Factors |
|---|---|---|
| 63 | 1, 3, 7, 9, 21, 63 | 3, 7 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | 2, 3 |
| 64 | 1, 2, 4, 8, 16, 32, 64 | 2 |
| 54 | 1, 2, 3, 6, 9, 18, 27, 54 | 2, 3 |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | 2, 3 |
Practice Problems: Try These Yourself
- List all the factors of 63.
- What are the factor pairs of 63?
- Find the prime factorization of 63.
- Is 21 a factor of 63?
- What is the sum of all the factors of 63?
Common Mistakes and Misunderstandings
- Confusing factors with multiples.
- Forgetting to check all pairs up to the square root.
- Missing negative factor pairs (for advanced students).
Relation to Other Concepts
Understanding factors of 63 helps you solve problems about factors of 72 and factors of 36, and is key for LCM, prime numbers, and simplification of fractions in higher classes.
Classroom Tip
Remember: “If you can divide 63 by a number without getting a fraction, that number is a factor!” At Vedantu, teachers often use grids and factor trees for instant factor listing in live classes.
We explored factors of 63 — their meaning, list, pairs, prime factorization, and tips for fast checking. Keep practicing with Vedantu to master factors and apply them in more complex math topics!
For similar practice, visit: Factors of 36, Factors of 72, Prime Factorization, or explore the general method of factorizing any number.
FAQs on Factors of 63 Explained with Examples
1. What are the factors of 63?
The factors of 63 are 1, 3, 7, 9, 21, and 63. These are the positive integers that divide 63 exactly without leaving any remainder.
- 63 ÷ 1 = 63
- 63 ÷ 3 = 21
- 63 ÷ 7 = 9
- 63 ÷ 9 = 7
- 63 ÷ 21 = 3
- 63 ÷ 63 = 1
2. How do you find the factors of 63?
To find the factors of 63, divide 63 by natural numbers and check which ones give a remainder of zero.
- Start from 1 and go up to 63.
- If 63 ÷ number = whole number, it is a factor.
- Stop checking at √63 (about 7.9) and list pairs.
3. What is the prime factorization of 63?
The prime factorization of 63 is 3 × 3 × 7 or 3² × 7. This means 63 is expressed as a product of prime numbers.
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
4. Is 63 a prime or composite number?
The number 63 is a composite number because it has more than two factors. A prime number has exactly two factors (1 and itself), but 63 has six factors: 1, 3, 7, 9, 21, and 63.
5. How many factors does 63 have?
The number 63 has 6 positive factors. Using prime factorization 63 = 3² × 7¹, apply the formula for total factors:
- Add 1 to each exponent: (2 + 1)(1 + 1)
- Multiply: 3 × 2 = 6
6. What are the factor pairs of 63?
The factor pairs of 63 are (1, 63), (3, 21), and (7, 9). These pairs multiply together to give 63.
- 1 × 63 = 63
- 3 × 21 = 63
- 7 × 9 = 63
7. What are the common factors of 63 and 42?
The common factors of 63 and 42 are 1, 3, 7, and 21. First list the factors:
- Factors of 63: 1, 3, 7, 9, 21, 63
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
8. What is the greatest common factor (GCF) of 63 and 84?
The greatest common factor (GCF) of 63 and 84 is 21. Using prime factorization:
- 63 = 3² × 7
- 84 = 2² × 3 × 7
9. Is 63 a perfect square?
The number 63 is not a perfect square because it cannot be written as the square of a whole number. Since √63 ≈ 7.94 and is not an integer, 63 is not a perfect square.
10. What is the smallest and largest factor of 63?
The smallest factor of 63 is 1 and the largest factor is 63. Every positive number has 1 as its smallest factor and the number itself as its greatest factor, including 63.
