Factors of 16 | Prime Factorization of 16 - Explained Simply

Today we are going to present here Factor Tree of 16. The factor is the number that divides the original number. The factors of 16 are 1, 2, 4, 8 and 16 itself.

What are the factors of 16?

The factors of 16 are all the whole numbers that, when divided into 16, result in a whole number with no remainder. We can also think of them as the whole numbers that multiply together to give 16.

Factor Tree Method of 16: Explained Simply

There are a couple of systematic ways to find all the factors of a number. We'll explore both, as they can help confirm your results.

Systematic Division

This method involves testing each whole number, starting from 1, to see if it divides 16 exactly.

Start with 1: Every whole number has 1 as a factor.

16 ÷ 1 = 16 (remainder 0)

So, 1 and 16 are factors.

Test 2:

16 ÷ 2 = 8 (remainder 0)

So, 2 and 8 are factors.

Test 3:

16 ÷ 3 = 5 with a remainder of 1.

So, 3 is not a factor of 16.

Test 4:

16 ÷ 4 = 4 (remainder 0)

So, 4 is a factor. (Notice that 4 is paired with itself).

Continue (and know when to stop):

If we were to test 5, 16 ÷ 5 gives a remainder.

We can stop checking once the number we're testing becomes greater than the quotient (the answer to the division). In the case of 4 × 4 = 16, we found a factor paired with itself (4). Once you reach the square root of the number (which is 4 for 16), or pass it, you've found all unique factor pairs. We already found 8 and 16, which are larger than 4.

Summary of Factors from Method 1: By dividing, we found the factors are 1, 2, 4, 8 and 16.

Finding Multiplication Pairs

This method involves listing pairs of whole numbers that multiply together to give 16.

Start with 1:

1 × 16 = 16

This gives us factors 1 and 16. Remember, 16 is also a factors of 32.

Move to 2:

2 × 8 = 16

This gives us factors 2 and 8.

Move to 3:

Is there a whole number that, when multiplied by 3, gives 16? No. (3 × 5 = 15, 3 × 6 = 18).

So, 3 is not a factor.

Move to 4:

4 × 4 = 16 = 4^2

This gives us the factor 4. (We only list it once, even though it appears twice in the multiplication).

Know when to stop: Once you start repeating the factors you've already found (for example, if you tested 8, you'd find 8 x 2 = 16, but you already have 2 and 8), or when the number you are testing becomes greater than the square root of 16 (which is 4), you have found all unique pairs.

Summary of Factors from Method 2: By finding multiplication pairs, we identified the factors as 1, 2, 4, 8 and 16.

The Complete List of Factors of 16

Combining the results from both methods, we can confidently state the factors of 16.

The factors of 16 are: 1, 2, 4, 8, 16.

Properties of Factors of 16

Understanding the factors opens the door to other important mathematical concepts.

1. Number of Factors

The number 16 has 5 factors. (Count them: 1, 2, 4, 8, 16).

2. Prime Factors

A prime number is a whole number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11...).

A prime factor is a factor that is also a prime number.

Looking at the factors of 16 (1, 2, 4, 8, 16), the only number that is prime is 2.

Therefore, the only prime factor of 16 is 2.

3. Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. It's like finding the fundamental prime numbers that multiply together to make the original number.

For 16:

16 = 2 × 8

8 = 2 × 4

4 = 2 × 2

So, 16 = 2 × 2 × 2 × 2 = 2^4

This can be written in exponential form as 2⁴.

4. Proper Factors

Proper factors are all the factors of

a number except the number itself.

For 16, the proper factors are 1, 2, 4, 8.

Prime Factorization of 16

Let's apply both methods to find the prime factorization of 16.

Using the Factor Tree Method for 16:

Start with 16 at the top.

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Find two factors of 16. Let's pick 2 and 8 (since 2 × 8 = 16).

```

/ \

② 8

```

Circle the 2 because it's a prime number. Now we need to break down 8.

Find two factors of 8. Let's pick 2 and 4 (since 2 × 4 = 8).

```

/ \

② 8

/ \

② 4

```

Circle the 2 because it's a prime number. Now we need to break down 4.

Find two factors of 4. Let's pick 2 and 2 (since 2^2 = 2 × 2 = 4).

```

/ \

② 8

/ \

② 4

/ \

② ②

```

Circle both 2s because they are prime numbers. All branches have ended in prime numbers!

Collect all the circled prime numbers: 2, 2, 2, 2.

Write them as a product: 2 × 2 × 2 × 2

Using the Division Method for 16:

The smallest prime number that divides 16 evenly is 2.

2 | 16

Divide 16 by 2. The quotient is 8.

2 | 16

2 | 8

Now take the quotient, 8. The

smallest prime number that divides 8 evenly is 2.

2 | 16

2 | 8

2 | 4

Now take the quotient, 4. The smallest prime number that divides 4 evenly is 2.

2 | 16

2 | 8

2 | 4

2 | 2

Now take the quotient, 2. The smallest prime number that divides 2 evenly is 2. The quotient is 1.

2 | 16

2 | 8

2 | 4

2 | 2

We stop when the quotient is 1. Now, collect all the prime divisors on the left side: 2, 2, 2, 2.

Write them as a product: 2 × 2 × 2 × 2

The Final Prime Factorization of 16:

Both methods lead to the same unique set of prime factors: 2 × 2 × 2 × 2 = 2^4.

This can also be written using exponents for a more compact form: 2⁴ (read as "2 to the power of 4").

As we've seen with the number 16, its prime factorization is a simple yet powerful expression: 2^4 = 2 × 2 × 2 × 2 or 2⁴.