Factors of 18 | Prime Factorization of 18 - Explained Simply
Today we are going to present here Factor Tree of 18. The factor is the number that divides the original number. The factors of 18 are 1,2,3,6,9 and 18 itself.
What are the factors of 18?
The factors of 18 are all the whole numbers that, when divided into 18, result in a whole number with no remainder. We can also think of them as the whole numbers that multiply together to give 18.
Factor Tree Method of 18: Explained Simply
Let's apply our understanding to find all the positive factors of 18. We can do this systematically by checking each number starting from 1 up to the number itself (or slightly less, as we'll see).
Here's a step-by-step process:
Start with 1: Is 1 a factor of 18? Yes, 18 ÷ 1 = 18.
Factor pair: (1, 18)
Check 2: Is 2 a factor of 18? Yes, 18 ÷ 2 = 9.
Factor pair: (2, 9)
Check 3: Is 3 a factor of 18? Yes, 18 ÷ 3 = 6.
Factor pair: (3, 6)
Check 4: Is 4 a factor of 18? No, 18 ÷ 4 = 4 with a remainder of 2.
Check 5: Is 5 a factor of 18? No, 18 ÷ 5 = 3 with a remainder of 3.
Check 6: Is 6 a factor of 18? Yes, 18 ÷ 6 = 3.
We've already found 6 and 3 as a pair. This means we've essentially "met in the middle" of our factor pairs, or we've reached a point where we're just reversing previous pairs. We don't need to check further numbers like 9 or 18, as they are already part of a pair we found.
The Factor Pairs of 18 are:
1 × 18
2 × 9
3 × 6
Therefore, the positive factors of 18 are: 1, 2, 3, 6, 9, 18.
Properties and Types of Factors for 18
Now that we've listed them, let's explore some interesting characteristics of these factors.
Total Number of Factors: 18 has 6 positive factors.
Even and Odd Factors:
Even factors: 2, 6, 18
Odd factors: 1, 3, 9
Prime vs. Composite Factors:
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself.
A composite number is a whole number greater than 1 that has more than two factors.
Factors of 18 that are Prime:
2 (factors: 1, 2)
3 (factors: 1, 3)
Factors of 18 that are Composite:
6 (factors: 1, 2, 3, 6)
9 (factors: 1, 3, 9)
18 (factors: 1, 2, 3, 6, 9, 18)
Special Case: 1
The number 1 is neither prime nor composite.
Prime Factorization of 18
There are two main methods for finding the prime factorization: the Factor Tree method and the Division method (or Ladder method). Both will give you the same unique answer.
The Factor Tree Method of 18
The factor tree method is a visual way to break down a number into its prime factors.
Start with the number at the top.
Draw two branches stemming from the number.
Find any two factors (besides 1 and the number itself) that multiply to give the number. Write these at the end of the branches.
Check if each factor is prime:
If a factor is prime, circle it. This branch is complete.
If a factor is composite, draw two more branches from it and repeat step 3.
Continue until all branches end in a circled prime number.
Collect all the circled prime numbers and write them as a product. Use exponents for repeated factors.
Let's find the Prime Factorization of 18 using a Factor Tree:
```
18
/ \
2 9 <-- (2 is prime, circle it. 9 is composite.)
/ \
3 3 <-- (3 is prime, circle both 3s.)
```
Result: The circled prime numbers are 2, 3, and 3.
So, the prime factorization of 18 is 2 × 3 × 3.
Using exponents for repeated factors, we write this as: 2 × 3²
The Division Method (Ladder Method) of 18
The division method is a more systematic way to find prime factors, especially useful for larger numbers.
Start with the number you want to factor.
Divide the number by the smallest prime number that divides it evenly. Write the prime divisor on the left and the result below the original number.
Repeat step 2 with the new result. Continue dividing by the smallest possible prime number until you can no longer divide evenly.
If the new result can no longer be divided by the previous prime, try the next smallest prime number.
Continue until the result of the division is 1.
The prime factorization is the list of all the prime divisors you used on the left side.
Let's find the Prime Factorization of 18 using the Division Method:
```
2 | 18
-----
3 | 9
-----
3 | 3
-----
1
```
Explanation: Start with 18. The smallest prime number that divides 18 evenly is 2.
18 ÷ 2 = 9.
Now we have 9. The smallest prime number that divides 9 evenly is not 2 (9 is odd), so we try the next prime, 3.
9 ÷ 3 = 3.
Now we have 3. The smallest prime number that divides 3 evenly is 3.
3 ÷ 3 = 1.
We stop when we reach 1.
Result: The prime numbers on the left are 2, 3, and 3.
So, the prime factorization of 18 is 2 × 3 × 3.
Using exponents, this is: 2 × 3²
Notice that both methods yield the exact same prime factorization for 18! This highlights the uniqueness mentioned in the Fundamental Theorem of Arithmetic.