__Objective: __

At the end of the lesson, students should be able to express a number as a product of its prime factors and write it in index form.

__Prime factors__

A **Prime factor** is a factor that is a prime number. As we know, a prime number can only be divided by itself and 1. Therefore a prime number has exactly two factors these are one and the number. For example **5** is a prime number because its factors are only 1 and 5. Note that **1** is not a prime factor** because it has only one factor not two.** **8** is not a prime number because it has more than two factors; 1, 2, 4 and 8. However, among the factors of 8 (**1, 2, 4 and 8**), **2** is a prime number therefore 2 is a prime factor of 8.The factors of 18 are {**1, 2, 3, 6, 9 and 18**.} 2 and 3 are prime numbers therefore 2 and 3 are the prime factors of 18

__Expressing a number as a product of its prime factors __

A product of primes has only prime numbers in it multiplication. For example, 20 expressed as a product of its prime factors is 2 x 2 x 5 = 20

Now if a number is broken down into factors which are prime numbers then the number is said to have been expressed as a product of its prime factors.

Note that every composite number can be written as a product of its prime factors

When a number is given as a product of its prime factors, it is said to be** factorized completely. **A prime factor tree is a special diagram where you find the factors of a number until you can’t factorise any more. The result is all the prime factors of the number. For example, the prime factor tree of 48 is shown below.

__Example 1 __

A factor tree for the number **48 **will always give “**2**“, “**2**“, “**2**“,”**2**“, and “**3**” as the prime factors.

The number 48 can be written as a product of its prime factors by multiplying these five numbers together.

Writing **48** as a product of its prime factors gives **2 × 2 × 2 ×2 ×3**

To write the prime factor in exponential form, count how many of each prime number there is. The number will become your exponent. **48 **= **2**^{4}×**3**

__Example 2__

Using a factor tree, write the following numbers as a product of their prime factors.

a) 8 b) 30 c) 42

__Solution__

a) 8=2×2×2 b) 30=2×3×5 c) 42= 2×3×7 **prime factors**

=2^{3} **index form**

__Example 3__

Express the following numbers as a product of their prime factors.

a.210 b. 180 c.72 d. 6160

__Solution__

a. 210= 2 × 105

=2 × 3 × 35

=2 × 3 × 5 × 7 **(prime factors)**

b. 180= 2 × 90

= 2× 2× 45

= 2× 2× 3× 15

=2 ×2 ×3× 3× 5 **(prime factors)**

=2^{2}×3^{3}× 5 **(index notation)**

c. 72=2×36

=2 ×2× 18

=2 ×2× 2× 9

=2× 2× 2× 3× 3 **(prime factors)**

=2^{3} ×3^{2 }**(index notation)**

d. 6160=2×3080

=2× 2× 1540

=2× 2× 2× 770

=2× 2× 2× 2 ×385

=2 ×2× 2× 5 ×77

=2×2× 2× 2× 5× 7× 11**(prime factors)**

=2^{4} ×5× 7× 11**(index notation)**

Note: A factor tree grows by continually splitting numbers into their factors until only prime numbers remain.

Write your prime factors as a product in full first then use the index form where appropriate.

The answers can be written in any order, but we tend to put the smallest prime factors first.

__Division ladder __

When using a division ladder, begin dividing by the smallest prime factor (in this case 2). Divide only with numbers which will divide exactly i.e. numbers which are factors. Use only prime numbers, and divide until the answer is 1

S0 70= 2×5×7

__Example 1__

Express the following as a product of their prime factors.

a)60 b)130

__Solution__

a) 60

60=2×2×3×5

Lesson Content

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