Factors of 23 | Prime Factorization of 23 - Explained Simply

Today we are going to present here Factor Tree of 23. The factor is the number that divides the original number. The factors of 23 are 1 and 23 itself.

What are the factors of 23?

The factors of 23 are all the whole numbers that, when divided into 23, result in a whole number with no remainder. We can also think of them as the whole numbers that multiply together to give 23.

Factor Tree Method of 23: Explained Simply

Now, let's apply our understanding to find the factors of 23. We'll use a systematic approach, checking each whole number starting from 1.

Steps to Find Factors of 23:

Start with 1:

Is 1 a factor of 23? Yes, 23 ÷ 1 = 23.

Factors found so far: 1

Check 2:

Is 2 a factor of 23? No, 23 ÷ 2 = 11 with a remainder of 1.

Check 3:

Is 3 a factor of 23? No, 23 ÷ 3 = 7 with a remainder of 2.

Check 4:

Is 4 a factor of 23? No, 23 ÷ 4 = 5 with a remainder of 3.

Continue checking numbers: We can stop checking once we reach a number whose square is greater than 23 (or its square root). The square root of 23 is approximately 4.79. This means we only need to check numbers up to 4. Since we haven't found any factors between 1 and 23 itself, we know there won't be any more.

We checked 2, 3, and 4 and found no factors.

Consider the number itself:

Is 23 a factor of 23? Yes, 23 ÷ 23 = 1.

Factors found so far: 1, 23.

After systematically checking all possible whole numbers, we find that the only numbers that divide 23 evenly are 1 and 23.

Therefore, the factors of 23 are 1 and 23.

The Significance of 23's Factors: Introducing Prime Numbers

The fact that 23 only has two factors (1 and itself) is incredibly significant in mathematics. It means that 23 is a special kind of number called a prime number.

Prime Factorization of 23

Now, let's apply our knowledge to the number 23. Is 23 a composite number that needs to be broken down, or is it one of those special prime building blocks itself?

To find the prime factorization of 23, we need to determine if it's a prime number. We do this by trying to divide 23 by small prime numbers, starting from the smallest.

Is 23 divisible by 2?

No. 23 is an odd number (it doesn't end in 0, 2, 4, 6, or 8). So, 2 is not a factor of 23.

Is 23 divisible by 3?

To check for divisibility by 3, we add the digits of the number: $2 + 3 = 5$.

Since 5 is not divisible by 3, 23 is not divisible by 3.

Is 23 divisible by 5?

No. For a number to be divisible by 5, it must end in a 0 or a 5. 23 does not.

Is 23 divisible by 7?

7 \times 1 = 7

7 \times 2 = 14

7 \times 3 = 21

7 \times 4 = 28

23 falls between 21 and 28, so it's not perfectly divisible by 7.

Important Tip: When checking for prime factors, you only need to test prime numbers up to the square root of the number you are factoring. The square root of 23 is approximately 4.79. This means we only needed to test prime numbers less than or equal to 4.79, which are 2 and 3. Since we've already tested both and found they are not factors, we can stop.

Conclusion

Since 23 cannot be divided evenly by any prime number other than 1 and itself, 23 is a prime number.

Therefore, the prime factorization of 23 is simply 23 itself. It is its own prime building block.

In mathematical notation: 23 = 23