This week in math 10, we focused mainly on prime factorization. Prime factorization is important because after finding the prime factors of a number, it helps us to find the GCF (greatest common factor) and the LCM (lowest common multiple) of the number.

Prime factorization is the decomposition of a composite number into a product of smaller prime numbers. The prime factors of a number are all the prime numbers that, when multiplied together, equal the original number.

Every composite number (*a whole number which has more than two factors) can be expressed as a product of prime factors. Expressing a whole number as a product of prime factors is called prime factorization.

With smaller numbers; like 12, the prime factorization can be done mentally. However, for larger numbers, the use of a division table or tree diagram may be applied. (see examples below)

 

**NOTE :

1. Only prime numbers can be used as divisors when using a division table.
2. Any divisor may be used when using a tree diagram. (as long as each number is divided into the smallest possible prime number)

Division Tables:

1. First we divide the number by the smallest prime number which divides the number exactly. (**Refer to the divisibility rules for prime numbers at the bottom of post)

Ex: 48 is divisible by 2. To demonstrate this, we would place the divisible number (48) inside of a bracket and the prime number (2) used to divide the number, on the outside of the bracket.

2. We then divide the quotient again by the smallest or the next smallest prime number if it is not exactly divisible by the smallest prime number. We repeat the process again and again until the quotient becomes 1. Remember, we use only prime numbers to divide.

Ex: 24 is divisible by 2. To demonstrate this, we would place the divisible number (24) below the last and the according prime number (2) outside the second bracket.

3. Finally, after dividing the number into prime factors, we then multiply all the prime factors together. Remember, the product of the prime numbers must be equal to the number itself.

Ex: all the prime factors on the outside of the brackets will be multiplied together to equal the number itself. If there are multiple prime factors of the same number, the factors can be written using exponents as 2 to the power of 4 multiplied by 3. (2⁴ x 3)

Second example :

Tree Diagrams :

1. Consider the number as the root of the tree. In this case the root number is 48.

2. Write a pair of factors as the branches of the tree i.e., 2 × 24 = 48 or 6 x 8 =48

3. Further factorize each composite factor until each branch reaches the smallest possible prime factor.

We repeat the process again until we get the prime factors of all the composite factors. Once all of the prime factors have been calculated, we multiply them together to receive the prime factorization. Remember, the product of the prime numbers must be equal to the number itself.

Second Example :