Prime Factorization and Factor Pairs of 64
The concept of factors of 64 is essential in mathematics and helps students in exam preparation, quick problem-solving, and building strong number sense. Being able to list, identify, and use the factors of numbers like 64 is a must-have skill for school and competitive exams.
What Are the Factors of 64?
Factors of 64 are all the whole numbers that divide 64 without leaving any remainder. In simple words, a factor of 64 is a number you can multiply by another whole number to get 64. This idea is used in topics such as divisibility, prime factorization, highest common factor, and multiples.
List of All Factors of 64
Let’s start with the complete list. The positive factors of 64 are:
- 1
- 2
- 4
- 8
- 16
- 32
- 64
These are all the numbers you can use to exactly divide 64. You can also include negative factors: -1, -2, -4, -8, -16, -32, -64.
| Factor | Division Statement | Quotient |
|---|---|---|
| 1 | 64 ÷ 1 | 64 |
| 2 | 64 ÷ 2 | 32 |
| 4 | 64 ÷ 4 | 16 |
| 8 | 64 ÷ 8 | 8 |
| 16 | 64 ÷ 16 | 4 |
| 32 | 64 ÷ 32 | 2 |
| 64 | 64 ÷ 64 | 1 |
Factor Pairs of 64
A factor pair is a set of two whole numbers, which multiplied together make 64. Here are all the positive factor pairs of 64:
| Factor Pair | Multiplication |
|---|---|
| (1, 64) | 1 × 64 = 64 |
| (2, 32) | 2 × 32 = 64 |
| (4, 16) | 4 × 16 = 64 |
| (8, 8) | 8 × 8 = 64 |
Remember, you can also write negative factor pairs: (-1, -64), (-2, -32), (-4, -16), and (-8, -8) because multiplying two negative numbers results in a positive product.
Is 64 a Prime or Composite Number?
64 is a composite number because it has more than two factors. A prime number has exactly two factors (1 and itself). 64 has seven positive factors. So, 64 is not a prime number.
Prime Factorization of 64
The prime factorization of 64 means writing 64 as a product of only prime numbers. Here’s how to find it step by step:
1. 64 ÷ 2 = 32 (2 is prime)2. 32 ÷ 2 = 16
3. 16 ÷ 2 = 8
4. 8 ÷ 2 = 4
5. 4 ÷ 2 = 2
6. 2 ÷ 2 = 1
So, the prime factorization of 64 is 2 × 2 × 2 × 2 × 2 × 2 = 26.
How to Find Factors of 64
There are two main methods to find the factors of any number, including 64:
- Check by Division:
Start dividing 64 by natural numbers starting from 1.
If you get a whole number result (no remainder), then that divisor is a factor.
For 64: 64 ÷ 1, 64 ÷ 2, 64 ÷ 4, etc. up to 64 ÷ 64. - Use Prime Factorization:
Break down the number by dividing by small prime numbers.
Continue dividing till you reach 1.
List out all combinations of the prime factors to find every factor.
Solved Examples on Factors of 64
Let’s solve a few typical exam questions:
1. List all the factors of 64.Answer: 1, 2, 4, 8, 16, 32, 64.
2. Is 24 a factor of 64?
64 ÷ 24 = 2.666... (not a whole number), so: No, 24 is NOT a factor of 64.
3. Find the greatest factor of 64.
Answer: 64 itself (every number is a factor of itself).
4. Find the factor pairs where both numbers are equal.
Answer: (8, 8) because 8 × 8 = 64.
Quick Reference: Factors Comparison Table
| Number | Factors |
|---|---|
| 32 | 1, 2, 4, 8, 16, 32 |
| 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
| 64 | 1, 2, 4, 8, 16, 32, 64 |
Try These Yourself
- Write all negative factors of 64.
- Check if 8 is both a factor and a multiple of 64.
- List all factor pairs of 32.
- Find the sum of all factors of 64.
Common Mistakes About Factors of 64
- Confusing factors with multiples (remember: multiples are greater than or equal to the number).
- Missing factor pairs like (8,8).
- Forgetting negative factors in some contexts.
Relation to Other Concepts
The factors of 64 are closely linked with concepts like prime factorization, multiples, HCF and LCM, and divisibility rules. Understanding factors helps you solve questions quickly in topics like number systems and algebra.
Tricks to Remember the Factors of 64
64 is a power of 2. If you know 2 × 2 × 2 × 2 × 2 × 2 is 64, you can quickly find all factors: they will all be powers of 2. Keep halving from 64 down to 1 to get all positive factors.
We learned about factors of 64—what they are, how to find them, prime factorization, and solved examples. To master these math basics and more, keep practicing on Vedantu’s interactive platform and check related topics:
FAQs on What Are the Factors of 64?
1. What are the factors of 64?
The factors of 64 are the numbers that divide 64 without leaving a remainder. These are: 1, 2, 4, 8, 16, 32, and 64. They can also be negative: -1, -2, -4, -8, -16, -32, and -64.
2. How many factors does 64 have?
64 has a total of 7 positive factors (1, 2, 4, 8, 16, 32, 64) and 7 negative factors (-1, -2, -4, -8, -16, -32, -64), for a total of 14 factors.
3. What are the prime factors of 64?
The prime factorization of 64 is 26. This means that the only prime factor of 64 is 2, and it appears six times in the prime factorization.
4. What are the factor pairs of 64?
The factor pairs of 64 are pairs of numbers that multiply to give 64. These include both positive and negative pairs: (1, 64), (2, 32), (4, 16), (8, 8), (-1, -64), (-2, -32), (-4, -16), (-8, -8).
5. Is 24 a factor of 64?
No, 24 is not a factor of 64 because 64 divided by 24 leaves a remainder.
6. What is the greatest factor of 64?
The greatest factor of 64 is 64 itself.
7. What is the prime factorization of 64?
The prime factorization of 64 is 2 x 2 x 2 x 2 x 2 x 2, or 26.
8. How can I use a factor tree to find the factors of 64?
A factor tree visually represents the prime factorization. Start with 64. Break it down into two factors (e.g., 8 x 8). Continue breaking down composite numbers until you only have prime numbers left. The prime numbers at the end of the branches are the prime factors of 64.
9. What is the difference between factors and multiples?
Factors are numbers that divide a given number exactly. Multiples are numbers obtained by multiplying a given number by other whole numbers. For example, the factors of 64 are 1, 2, 4, 8, 16, 32, and 64, while the multiples of 64 are 64, 128, 192, and so on.
10. Is 64 a perfect square?
Yes, 64 is a perfect square because it is the square of an integer (8 x 8 = 64).
11. Is 64 a perfect cube?
Yes, 64 is also a perfect cube because it is the cube of an integer (4 x 4 x 4 = 64).
