What Are the Factors and Factor Pairs of 63?
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: Definition, List, and Prime Factorization
1. What are the factors of 63?
The factors of 63 are the numbers that divide 63 exactly without leaving a remainder. These are: 1, 3, 7, 9, 21, and 63. We can find these by systematically dividing 63 by each whole number until we reach 63 itself.
2. What is the prime factorization of 63?
The prime factorization of 63 expresses it as a product of prime numbers. It is 3 x 3 x 7, or 32 x 7. This means 63 can be broken down into only these prime factors.
3. How do I find the factors of 63 quickly?
To find factors quickly, start by dividing 63 by the smallest prime numbers (2, 3, 5, 7, etc.). If the result is a whole number, both the divisor and the quotient are factors. For example: 63 ÷ 3 = 21, so 3 and 21 are factors. Continue this process until you've found all the factors.
4. What are the factor pairs of 63?
The factor pairs of 63 are pairs of numbers that multiply to 63. These are: (1, 63), (3, 21), and (7, 9). Remember that each pair can also be expressed with negative numbers: (-1, -63), (-3, -21), and (-7, -9).
5. Is 63 a prime or composite number?
63 is a composite number because it has more than two factors (1 and itself). Prime numbers only have two factors.
6. What are the negative factors of 63?
Along with the positive factors, 63 also has negative factors: -1, -3, -7, -9, -21, and -63. These are the opposites of the positive factors.
7. How is a factor tree used to find the factors of 63?
A factor tree is a visual way to find the prime factors. Start with 63. Since it's divisible by 3, branch it into 3 and 21. Then break down 21 into 3 and 7. You end up with the prime factors 3, 3, and 7. All other factors are combinations of these primes.
8. What are some common factors of 63 and other numbers?
To find common factors, list the factors of 63 and another number. Then identify the numbers that appear in both lists. For instance, 63 and 21 share common factors of 1, 3, 7 and 21.
9. How can I use factors of 63 to solve problems?
Understanding factors is crucial for simplifying fractions, finding the greatest common factor (GCF), and the least common multiple (LCM) of numbers—concepts vital in many math problems. Factors of 63 help in solving problems relating to divisibility and factorization.
10. What is the sum of all the factors of 63?
To find the sum, add all the factors together: 1 + 3 + 7 + 9 + 21 + 63 = 104. This is the sum of all the positive factors of 63.
11. Explain the difference between factors and multiples of 63.
Factors of 63 are numbers that divide 63 without a remainder (e.g., 1, 3, 7, 9, 21, 63). Multiples of 63 are numbers obtained by multiplying 63 by whole numbers (e.g., 63, 126, 189, ...).
12. Can you provide examples of real-world applications of factors of 63?
While there aren't many direct, obvious real-world examples specifically involving the factors of 63, the concept of factors is widely applicable. For instance, if you have 63 candies to divide equally among friends, knowing the factors helps determine how many groups you can make. The broader concept of factorization is used in various fields such as cryptography and coding.
