Factors of 32 | Prime Factorization of 32 - Explained Simply
Today we are going to present here Factor Tree of 32. The factor is the number that divides the original number. The Factors of 32 are 1, 2, 4, 8, 16 and 32 itself.
What are the factors of 32?
To find the factors of 32, we look for all integers that divide 32 evenly, meaning without a remainder. We can systematically check integers starting from 1.
Factor Tree Method of 32: Explained Simply
$$1 \times 32 = 32$$
So, 1 and 32 are factors.
$$2 \times 16 = 32$$
So, 2 and 16 are factors.
3 does not divide 32 evenly. \(32 \div 3 = 10\) with a remainder of 2.
$$4 \times 8 = 32$$
So, 4 and 8 are factors.
5 does not divide 32 evenly. \(32 \div 5 = 6\) with a remainder of 2.
We continue checking up to the square root of 32. The square root of 32 is approximately 5.66. Since we've already checked 1, 2, and 4, and their corresponding pairs (32, 16, 8), we have found all unique factors. The next integer to check would be 6, but we can see that if we were to find a factor larger than 5.66, its pair would be smaller than 5.66 and would have already been found.
Collecting all the unique factors we found in ascending order:
1, 2, 4, 8, 16, 32
Therefore, the factors of 32 are 1, 2, 4, 8, 16, and 32.
Prime Factorization of 32
There are a couple of common methods to find the prime factorization of a number: the Factor Tree method and the Repeated Division (or Ladder) method. Both will lead you to the same unique answer. We'll explore both for 32.
Prime Factorization of 32 Using the Factor Tree Method
The factor tree method is a visual way to break down a number. You start with the number at the top and branch out its factors until all branches end in prime numbers.
Let's find the Prime Factorization of 32:
Start with 32 at the top:
```
32
```
Find two factors of 32. It's usually easiest to start with the smallest prime number, 2, if the number is even. 32 is even, so we can divide it by 2.
32 = 2 × 16
```
32
/ \
2 16
```
Check if the factors are prime.
Is 2 a prime number?
Yes! Circle it to show we're done with that branch.
Is 16 a prime number?
No, it's composite. We need to break it down further.
```
32
/ \
(2) 16
```
Continue with the composite number (16). Find two factors of 16. Again, 16 is even, so divide by 2.
16 = 2 × 8
```
32
/ \
(2) 16
/ \
2 8
```
Check factors.
Is 2 prime? Yes! Circle it.
Is 8 prime? No, it's composite. Break it down.
```
32
/ \
(2) 16
/ \
(2) 8
```
Continue with 8. Find two factors of 8. 8 is even, so divide by 2.
8 = 2 × 4
```
32
/ \
(2) 16
/ \
(2) 8
/ \
2 4
```
Check factors.
Is 2 prime? Yes! Circle it.
Is 4 prime? No, it's composite. Break it down.
```
32
/ \
(2) 16
/ \
(2) 8
/ \
(2) 4
```
Continue with 4. Find two factors of 4. 4 is even, so divide by 2.
4 = 2 × 2
```
32
/ \
(2) 16
/ \
(2) 8
/ \
(2) 4
/ \
2 2
```
Check factors.
Are both 2s prime? Yes! Circle them. Now all branches end in circled prime numbers.
```
32
/ \
(2) 16
/ \
(2) 8
/ \
(2) 4
/ \
(2) (2)
```
Collect all the circled prime numbers.
The prime factors of 32 are: 2, 2, 2, 2, 2.
Prime Factorization of 32 Using the Repeated Division Method
This method involves repeatedly dividing the number by the smallest possible prime number until the result is 1.
Let's find the Prime Factorization of 32:
Start with 32. Divide it by the smallest prime number that divides it evenly.
32 ÷ 2 = 16
Take the result (16). Divide it by the smallest prime number that divides it evenly.
16 ÷ 2 = 8
Take the result (8). Divide it by the smallest prime number that divides it evenly.
8 ÷ 2 = 4
Take the result (4). Divide it by the smallest prime number that divides it evenly.
4 ÷ 2 = 2
Take the result (2). Divide it by the smallest prime number that divides it evenly.
2 ÷ 2 = 1
Stop when you reach 1. The prime factors are all the numbers you divided by.
The prime factors of 32 are: 2, 2, 2, 2, 2.
Conclusion:
Both methods yield the same prime factors. To write the prime factorization, we express the original number as a product of these prime factors.
Long Form: 32 = 2 × 2 × 2 × 2 × 2
For numbers with repeated prime factors, we use exponential notation for a more concise form.
Exponential Form: Since the number 2 appears 5 times, we can write it as 2 raised to the power of 5.
32 = 2⁵
This means "2 multiplied by itself 5 times."
The prime factorization of 32 is 2 × 2 × 2 × 2 × 2, or 2⁵. This is the unique set of prime numbers that multiply together to form 32.