Factors of 36 | Prime Factorization of 36 - Explained Simply

Today we are going to present here Factor Tree of 36. The factor is the number that divides the original number. The Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36 itself.

What are the factors of 36?

To find the factors of 36, we look for all integers that divide 36 evenly, meaning without a remainder. We can systematically check integers starting from 1.

Factor Tree Method of 36: Explained Simply

There are a few systematic ways to find all the factors of any number. Let's apply them to 36.

Method 1: The Division Method (Systematic Checking)

This method involves checking each whole number, starting from 1, to see if it divides 36 evenly.

Start with 1:

Is 36 divisible by 1? Yes, 36 ÷ 1 = 36.

Factors found: 1, 36

Check 2:

Is 36 divisible by 2? Yes, 36 ÷ 2 = 18.

Factors found: 1, 2, 18, 36

Check 3:

Is 36 divisible by 3? Yes, 36 ÷ 3 = 12.

Factors found: 1, 2, 3, 12, 18, 36

Check 4:

Is 36 divisible by 4? Yes, 36 ÷ 4 = 9.

Factors found: 1, 2, 3, 4, 9, 12, 18, 36

Check 5:

Is 36 divisible by 5? No, it leaves a remainder (36 ÷ 5 = 7 with 1 remainder). So, 5 is NOT a factor.

Check 6:

Is 36 divisible by 6? Yes, 36 ÷ 6 = 6.

Factors found: 1, 2, 3, 4, 6, 9, 12, 18, 36 (Note: we only list 6 once, even though it appears twice in the multiplication 6x6).

When to Stop? You can stop checking numbers once you reach a factor that you've already found in a pair, or when the number you are checking is greater than its corresponding factor. In our case, after 6, the next number to check would be 7, but its corresponding pair would be 36/7, which is not a whole number. The next factor pair found was 4 and 9. When we get to checking 9, we've already found 9. This means we've found all the factors!

Method 2: Multiplication Pairs

This method focuses on finding pairs of numbers that multiply together to give 36.

1 × 36 = 36

2 × 18 = 36

3 × 12 = 36

4 × 9 = 36

6 × 6 = 36

Once you start seeing pairs where the first number is equal to or greater than the second number (like 6x6, or if we kept going and got to 9x4), you know you've found all the unique pairs.

Listing All the Factors of 36

Combining the results from either method, we can now confidently list all the factors of 36 in ascending order:

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Notice how factors come in pairs (1 & 36, 2 & 18, 3 & 12, 4 & 9).

The factor 6 is paired with itself, which is characteristic of a square number.

The number 36 is an excellent choice for exploring factors because:

It's a "Perfect" Square: 36 is the result of multiplying a number by itself (6 × 6 = 36). Square numbers often have an interesting distribution of factors.

It Has Many Factors: Unlike a prime number (which only has two factors: 1 and itself), 36 has a good variety of factors, making it perfect for demonstrating different methods of finding them.

Practical Applications: 36 frequently appears in real-world scenarios, such as counting items in dozens (3 dozens), arranging objects, or in time calculations (36 minutes).

Prime Factorization of 36

The factor tree method is a visual and intuitive way to find prime factors.

Step-by-Step for 36:

Start with the number at the top: Write down 36.

```

Find any two factors: Choose any pair of factors for 36 (other than 1 and 36). Let's pick 6 and 6. Draw branches connecting 36 to 6 and 6.

```

/ \

6 6

```

Break down non-prime factors: Look at the numbers at the ends of the branches. Are they prime?

No, 6 is not prime (it has factors 2 and 3).

So, break down each 6 into its factors. Let's use 2 and 3.

```

/ \

6 6

/ \ / \

2 3 2 3

```

Circle the prime numbers: As soon as you reach a prime number at the end of a branch, circle it. These are your prime factors.

```

/ \

6 6

/ \ / \

(2) (3) (2) (3)

```

All the numbers at the very bottom of the tree are now prime (2 and 3 are prime).

Collect all the circled prime factors:

From the tree, we have: 2, 3, 2, 3

Write the prime factorization as a product:

36 = 2 × 3 × 2 × 3

Write in exponential form (optional, but standard):

Group identical prime factors together and use exponents.

36 = 2² × 3² (read as "2 squared times 3 squared")

What if we started with different factors?

Let's try starting with 4 and 9:

```

/ \

4 9

/ \ / \

(2) (2) (3) (3)

```

Notice the circled primes are still 2, 2, 3, 3! The result is the same: 36 = 2² × 3². This illustrates the Fundamental Theorem of Arithmetic perfectly.

The Division Method (Ladder Method)

The division method is a more systematic approach, often called the "ladder method."

Step-by-Step for 36:

Start with the number: Write down 36.

Divide by the smallest prime factor: Find the smallest prime number that divides evenly into 36. This is 2.

36 ÷ 2 = 18

Continue with the result: Take the result (18) and find the smallest prime number that divides evenly into it. Again, it's 2.

18 ÷ 2 = 9

Repeat until the result is 1: Now take 9. The smallest prime that divides into 9 is 3.

9 ÷ 3 = 3

Final step: Take 3. The smallest prime that divides into 3 is 3 itself.

3 ÷ 3 = 1

You stop when the result of your division is 1.

Visually, it looks like this (imagine lines creating a ladder):

```

2 | 36

2 | 18

3 | 9

3 | 3

| 1

```

Collect all the prime divisors: The prime factors are all the numbers you divided by on the left side of your ladder.

These are: 2, 2, 3, 3

Write the prime factorization as a product:

36 = 2 × 2 × 3 × 3

Write in exponential form:

36 = 2² × 3²

Both methods consistently give us the same prime factorization for 36:

36 = 2 × 2 × 3 × 3

Or, more concisely, using exponents:

36 = 2² × 3²

This means that the number 36 is built from two "2"s and two "3"s multiplied together. There's no other combination of prime numbers that will multiply to give you exactly 36. This unique "prime fingerprint" is what makes prime factorization so powerful and fundamental in number theory.