Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. Including 1 and 36 itself, there are 9 distinct factors for 36.
The prime factors of 36 are 2, 3, and its factor pairs are (1, 36), (2, 18), (3, 12), (4, 9), (6, 6). We've put this below in a table for easy sharing.
| Factors of 36: | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| Prime Factors of 36: | 2, 3 |
| Factor Pairs of 36: | (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) |
How to calculate factors?
To be a factor of 36, a number must divide 36 exactly, leaving no remainder. In other words, when 36 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 36 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 36 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 36:
| Divisor | Is it a factor of 36? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 36 = 36 |
| 2 | Yes, 36 is an even number so it's divisible by 2. | 2 × 18 = 36 |
| 3 | Yes, the sum of its digits (9) is divisible by 3. | 3 × 12 = 36 |
| 4 | Yes, the last two digits (36) form a number divisible by 4. | 4 × 9 = 36 |
| 5 | No, last digit is 6, so not divisible by 5. | - |
| 6 | Yes, 36 is divisible by both 2 and 3. | 6 × 6 = 36 |
| 7 | No, 36 divided by 7 leaves a remainder of 1. | - |
| 8 | No, the last three digits (36) do not form a number divisible by 8. | - |
| 9 | Yes, the sum of its digits (9) is divisible by 9. | 9 × 4 = 36 |
| 10 | No, last digit is 6, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (3) is not divisible by 11. | - |
| 12 | Yes, 36 is divisible by both 3 and 4. | 12 × 3 = 36 |
| ... | 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 36
You start by dividing 36 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 \( \sqrt36 \)
| Prime Number | Is it a factor of 36? | Verification |
|---|---|---|
| 2 | Yes, 36 is divisible by 2. | 36 ÷ 2 = 18, R0 |
| 2 | Yes, the result 18 is divisible by 2. | 18 ÷ 2 = 9, R0 |
| 2 | No, the result 9 is not divisible by 2. | - |
| 3 | Yes, 9 is divisible by 3. | 9 ÷ 3 = 3, R0 |
| 3 | Yes, the result 3 is divisible by 3. | 3 ÷ 3 = 1, R0 |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 36 by performing successive divisions. This method involves dividing 36 by every integer from 1 up to 36 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 36, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 36 ÷ 1 = 36 |
| 2 | 36 ÷ 2 = 18 |
| 3 | 36 ÷ 3 = 12 |
| 4 | 36 ÷ 4 = 9 |
| 6 | 36 ÷ 6 = 6 |
| 9 | 36 ÷ 9 = 4 |
| 12 | 36 ÷ 12 = 3 |
| 18 | 36 ÷ 18 = 2 |
| 36 | 36 ÷ 36 = 1 |
Using the division method, we calculated that factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
Factor Tree of 36
The factor tree of 36 shows the step-by-step breakdown of 36 into its prime factors. Each branch of the tree represents a division of 36 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 36 and highlights the structure of its prime factorization.
| 36 | |||
| | | \ | ||
| 2 | 18 | ||
| | | \ | ||
| 2 | 9 | ||
| | | \ | ||
| 3 | 3 |
Factor Pairs of 36 (Visualization)
Factor pairs of 36 are sets of two numbers that, when multiplied together, result in 36. Factor pairs are symmetric and mirror around the square root of 36, such as (1, 36) and (36, 1), and can be both positive and negative pairs as long as their product equals 36.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -36) | (1, 36) |
| (-2, -18) | (2, 18) |
| (-3, -12) | (3, 12) |
| (-4, -9) | (4, 9) |
| (-6, -6) | (6, 6) |
All factor pairs of 36 are (1, 36), (2, 18), (3, 12), (4, 9), (6, 6), (-1, -36), (-2, -18), (-3, -12), (-4, -9), (-6, -6).
Why Should I Care About Factors of 36?
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:
- Setting Up Chairs: You have 36 chairs and need to arrange them into rows for an event. If you have 3 rows, you’ll need 12 chairs per row because 3 × 12 = 36.
- Distributing Supplies: You have 36 school supplies and want to divide them equally among 6 classrooms. Each classroom will get 6 supplies because 6 × 6 = 36.
- Creating Posters: You have 36 posters to hang up. If you want to divide the task among 4 people, each person will hang 9 posters because 4 × 9 = 36.
- Arranging Tables: You have 36 tables to arrange for a party. If each row has 4 tables, you’ll create 9 rows because 4 × 9 = 36.
- Splitting a Pizza: You have 36 slices of pizza. If you split the pizza between 1 friends, each friend will get 36 slices because 1 × 36 = 36.
