__Expressing numbers as a product of prime numbers__

To express a number as a product of prime numbers, we draw a 2-column table and placing the number at the right column. The left column will contain the prime numbers we want to use to divide.

Always start with the first prime number (2), if 2 the number is nit divisible by 2 we try 3, 5, 7 and so on.

__Examples__

Express the numbers below as a product of prime factors

1. 96

__Solution__

2 | 96 |

2 | 48 |

2 | 24 |

2 | 12 |

2 | 6 |

3 | 3 |

1 |

96 = 2 x 2 x 2 x 2 x 2 x 3

We keep dividing by 2 and writing the answer till it can no longer divide our answer, then we try 3

We can also express our answer in index form, i.e.

2 x 2 x 2 x 2 x 2 x 3 will be 2^{5} x 3. The power 5 is the number of times 2 appears in the multiplication.

2. 135 leaving your answer in index form

__Solution__

3 | 135 |

3 | 45 |

3 | 15 |

5 | 5 |

1 |

135 = 3 x 3 x 3 x 5

= 3^{3} x 5

3. 150

__Solution__

2 | 150 |

3 | 75 |

5 | 25 |

5 | 5 |

1 |

150 = 2 x 3 x 5 x 5

= 2 x 3 x 5^{2}.

** EXERCISE C**

Express the numbers below as a product of prime factors in index form.

1. 36

2. 57

3. 120

4. 135

5. 512

6. 840

7. 1700

8. 2250

9. 3000

10. 888

** Common Factors**

Common factors exist between two or more numbers. When their factors are listed, the common ones are the common factors.

** **

__Example__Find the common factors of the numbers below:

1. 24 and 36.

__Solution__

We first list the factors of each of the given number

*24 – 1, 2, 3, 4, 6, 8, 12, 24*

*36 – 1, 2, 3, 4, 6, 9, 12, 18, 36*

From the factors listed, the ones that are common to the two numbers are 1, 2, 3, 4, 6, and 12. They are the common factors.

2. 63 and 108

__Solution__

63 *– 1, 3, 7, 9, 21, 63*

108 – *1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108*

Common factors are 1, 3 and 9

3. 70, 105 and 35

__Solution__

60 – *1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60*

105 – *1, 3, 5, 7, 15, 21, 35, 105*

35 – *1, 5, 7, 35*

Common factors are 1 and 5.

From the examples above, we will observe that 1 is always a common factor.

__Exercise D__

Find the common factors of the numbers below:

1. 20 and 30

2. 45, 30 and 75

3. 40, 50 and 60

4. 18, 24, 12 and 36

5. 56, 21 and 35

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