Factors of 64 are 1, 2, 4, 8, 16, 32, 64. Including 1 and 64 itself, there are 7 distinct factors for 64.
The prime factors of 64 are 2, and its factor pairs are (1, 64), (2, 32), (4, 16), (8, 8). We've put this below in a table for easy sharing.
| Factors of 64: | 1, 2, 4, 8, 16, 32, 64 |
| Prime Factors of 64: | 2 |
| Factor Pairs of 64: | (1, 64), (2, 32), (4, 16), (8, 8) |
How to calculate factors?
To be a factor of 64, a number must divide 64 exactly, leaving no remainder. In other words, when 64 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 64 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 64 using four methods: basic factorization, prime factorization, the division method, or using GCD and LCM. We also include a detailed analysis of factor pairs and a factor tree to illustrate the breakdown.
Method 1: Basic Factorization
Basic Factorization is a method to find the factors of a number by systematically testing each whole number from 2 up to the number itself to see which ones divides with zero remainder (evenly). The process is somewhat time consuming if a number is high, that's why you should master divisibility rules, to make the process faster.
Here's the breakdown for 64:
| Divisor | Is it a factor of 64? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 64 = 64 |
| 2 | Yes, 64 is an even number so it's divisible by 2. | 2 × 32 = 64 |
| 3 | No, the sum of its digits (10) is not divisible by 3. | - |
| 4 | Yes, the last two digits (64) form a number divisible by 4. | 4 × 16 = 64 |
| 5 | No, last digit is 4, so not divisible by 5. | - |
| 6 | No, 64 is not divisible by both 2 and 3. | - |
| 7 | No, 64 divided by 7 leaves a remainder of 1. | - |
| 8 | Yes, the last three digits (64) form a number divisible by 8. | 8 × 8 = 64 |
| 9 | No, the sum of its digits (10) is not divisible by 9. | - |
| 10 | No, last digit is 4, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (2) is not divisible by 11. | - |
| 12 | No, 64 is not divisible by both 3 and 4. | - |
| ... | continue with all the other numbers. |
Method 2: Prime Factorization
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This means a prime number cannot be formed by multiplying two smaller natural numbers. For example:
- 2 is a prime number because its only divisors are 1 and 2.
- 3 is prime for the same reason—it can only be divided evenly by 1 and 3.
- 4 is not prime because it can be divided by 1, 2, and 4.
- 5, 7, 11, and 13 are also prime numbers.
Prime numbers are fundamental in mathematics because they are the "building blocks" of whole numbers. Any natural number greater than 1 can be expressed as a product of prime numbers, which is known as its prime factorization.
How to do prime factorization of 64
You start by dividing 64 to each prime number, multiple times, until the remainder is 0. Then you move on to the next prime number. To save time, you should test with up to \( \sqrt64 \)
| Prime Number | Is it a factor of 64? | Verification |
|---|---|---|
| 2 | Yes, 64 is divisible by 2. | 64 ÷ 2 = 32, R0 |
| 2 | Yes, the result 32 is divisible by 2. | 32 ÷ 2 = 16, R0 |
| 2 | Yes, the result 16 is divisible by 2. | 16 ÷ 2 = 8, R0 |
| 2 | Yes, the result 8 is divisible by 2. | 8 ÷ 2 = 4, R0 |
| 2 | Yes, the result 4 is divisible by 2. | 4 ÷ 2 = 2, R0 |
| 2 | Yes, the result 2 is divisible by 2. | 2 ÷ 2 = 1, R0 |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 64 by performing successive divisions. This method involves dividing 64 by every integer from 1 up to 64 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 64, and only kept those that do:
| Divisor | Verification |
|---|---|
| 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 |
Using the division method, we calculated that factors of 64 are 1, 2, 4, 8, 16, 32, 64.
Factor Tree of 64
The factor tree of 64 shows the step-by-step breakdown of 64 into its prime factors. Each branch of the tree represents a division of 64 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 64 and highlights the structure of its prime factorization.
| 64 | |||||
| | | \ | ||||
| 2 | 32 | ||||
| | | \ | ||||
| 2 | 16 | ||||
| | | \ | ||||
| 2 | 8 | ||||
| | | \ | ||||
| 2 | 4 | ||||
| | | \ | ||||
| 2 | 2 |
Factor Pairs of 64 (Visualization)
Factor pairs of 64 are sets of two numbers that, when multiplied together, result in 64. Factor pairs are symmetric and mirror around the square root of 64, such as (1, 64) and (64, 1), and can be both positive and negative pairs as long as their product equals 64.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -64) | (1, 64) |
| (-2, -32) | (2, 32) |
| (-4, -16) | (4, 16) |
| (-8, -8) | (8, 8) |
All factor pairs of 64 are (1, 64), (2, 32), (4, 16), (8, 8), (-1, -64), (-2, -32), (-4, -16), (-8, -8).
Why Should I Care About Factors of 64?
Turns out, factors aren’t just about boring math equations—they’re like secret superpowers hiding inside numbers! Knowing them can help you split things up, share with friends, or even spot hidden patterns. Want to know how? Check out these real-life examples that show just how cool factors really are:
- Handing Out Invitations: You have 64 invitations to give out. If each person gets 4 invitations, you will need 16 people to distribute them because 4 × 16 = 64.
- Laying Bricks: You have 64 bricks to lay for a pathway. If each row requires 8 bricks, you’ll create 8 rows because 8 × 8 = 64.
- Planting Flowers: You have 64 flower seeds to plant in a garden. If each row contains 4 seeds, you’ll plant 16 rows because 4 × 16 = 64.
- Stacking Books: You have 64 books to stack in a library. If each stack has 1 books, you’ll have 64 stacks because 1 × 64 = 64.
- Filling Party Favors: You have 64 items to put in party favor bags. If each bag holds 8 items, you’ll fill 8 bags because 8 × 8 = 64.
