Factors of 15 | Prime Factorization of 15 - Explained Simply

Today we are going to present here Factor Tree of 15. The factor is the number that divides the original number. The factors of 15 are 1, 3, 5 and 15 itself.

What are the factors of 15?

We want to find all the whole numbers that can divide 15 without leaving any remainder.

Factor Tree Method of 15: Explained Simply

Finding the factors of a number like 15 is straightforward. We can use a couple of systematic methods to ensure we don't miss any.

Method 1: Systematic Division

This method involves checking each whole number, starting from 1, to see if it divides 15 evenly.

Start with 1:

15 ÷ 1 = 15.

So, 1 and 15 are factors.

Check 2:

15 ÷ 2 = 7 with a remainder of 1.

So, 2 is not a factor.

Check 3:

15 ÷ 3 = 5.

So, 3 and 5 are factors.

Check 4:

15 ÷ 4 = 3 with a remainder of 3.

So, 4 is not a factor.

Check 5:

We've already found 5 as a factor when we divided by 3. This tells us we've found all the pairs, or we can stop when the number we are checking is greater than the result of the division (e.g., 15/3 = 5, 15/4 = 3.75, 15/5 = 3. Once the divisor is larger than the quotient, we've covered all unique pairs).

By systematically checking, we've identified all the factors.

Method 2: Multiplication Pairs

This method involves thinking about which pairs of whole numbers multiply together to give 15.

Start with 1:

What number multiplied by 1 gives 15?

1 x 15 = 15.

Factors found: 1, 15.

Move 2:

Can 2 be multiplied by a whole number to get 15?

No (2 x 7 = 14, 2 x 8 = 16).

Move 3:

Can 3 be multiplied by a whole number to get 15?

Yes! 3 x 5 = 15.

Factors found: 3, 5.

Move 4:

Can 4 be multiplied by a whole number to get 15?

No (4 x 3 = 12, 4 x 4 = 16).

Move 5:

We've already found 5 in our previous pair (3 x 5). This indicates we have found all unique factor pairs. When you start repeating factors, you know you're done.

The Factors of 15: A Clear List

3|15

5|5

Based on both methods, we can confidently list all the factors of 15.

The factors of 15 are 1, 3, 5, 15.

There are 4 factors for the number 15.

The Fundamental Theorem of Arithmetic

This powerful mathematical theorem is the backbone of prime factorization. In simple terms, it states:

Every composite number can be broken down into a product of prime numbers.

This prime factorization is unique, no matter how you start the process (the order of the prime factors might change, but the set of primes themselves will always be the same).

This uniqueness is what makes prime factorization so reliable and useful.

Prime Factorization of 15

Here we discussed Step-by-Step Prime Factorizing the Number 15

Now, let's put our knowledge into practice and find the prime factors of 15. We'll use two common methods that achieve the same result.

The Division Method (Systematic Division by Primes)

Start with the number: Our number is 15.

Find the smallest prime factor:

Is 15 divisible by 2?

No, because 15 is an odd number.

Is 15 divisible by 3?

Yes! 15 \ 3 = 5.

We've found our first prime factor 3. The remaining number is 5.

Check if the result is prime: Is 5 a prime number? Yes, it is (only divisible by 1 and 5).

Stop: Since 5 is prime, we stop the process.

Write the prime factorization: The prime factors are the numbers we divided by and the final prime number.

Therefore, the prime factorization of 15 is {3 × 5}.

The Factor Tree Method (Visual Approach)

The factor tree is a visual way to break down a number into its prime factors.

Start with the number at the top:

```

Find any two factors of the number (other than 1 and itself):

For 15, we know that 3 × 5 = 15.

```

/ \

3 5

```

Check if the factors are prime:

Is 3 a prime number?

Yes. Circle it to show it's a prime branch end.

Is 5 a prime number?

Yes. Circle it.

```

/ \

③ ⑤

```

Since all branches end in prime numbers, the tree is complete.

Write the prime factorization: Multiply all the circled prime numbers together.

So, the prime factorization of 15 is {3 × 5}.

Both methods lead us to the same unique answer, confirming the Fundamental Theorem of Arithmetic!

Verifying Our Work

To double-check our prime factorization, simply multiply the prime factors together:

3 × 5 = 15

Since the product equals, our original number, our prime factorization is correct!