PRIME AND COMPOSITE NUMBERS, FACTORS |

**What is Factoring?**

To factor a number, we express it as a **product** of other **numbers** that **multiply** to give the original number. **Usually**, but not always, we use **whole numbers, called integers** for factoring.

For example, we can factor **12** as **2** × **6**, **3** × **4**, or **12** × **1**.

And we can factor **18** as **2** × **9**, **3** × **6**, or **18** × **1**

Rarely would we factor **18** as **½** × **36**, however, it might be useful in some cases.

**Factoring** is an extremely important mathematical technique, since it lets us break down large numbers and complicated algebraic expressions into a simplified **product** form, which helps us solve mathematical questions.

The one thing to always remember about factoring is that

**the product of the factors must equal the original number or expression**.

In other words, **if you multiply out the factors, you should get exactly what you started with**. Obviously, then, we HAVE TO KNOW OUR MULTIPLICATION TABLES (*oh no!!*) -- if we want to factor numbers correctly.

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**Prime and Composite Numbers and Their Factors**

** definitions:
**1) The

2) A ** prime number** is a whole number (integer) whose

So 2, 3, 5, 7, and 11 are prime numbers since 2 = 2 × 1, 3 = 3 × 1, etc.

Since any **even number** can be divided by **2**, it **is the only even prime** number.

All prime numbers bigger than 2 **are odd numbers**.

Here are all the prime numbers between 2 and 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

3) A ** composite number** can be expressed as a

Any even number bigger than 2 is a **composite number**.

4) The **prime factors **of a number are the **prime numbers** that are **factors** of the given number.

So the **prime factors** of **12** are **1, 2**, and **3** since **1**** ****×**** 2 ****×**** 2 ****×**** 3 **= **12**.

And, since **2 ****×**** 2** = 2², we could and should write the factors of 12 as **1 ×**** 2²****×**** 3**

**Example**: **6 and 3** or **9 and 2** are factors of 18, but if we want the **prime factors**,

we would have to write **6 **as 3 **×**** 2** and we'd make **9** = **3**** × ****3**,

because both **6 and 9 are composite** numbers.

**The prime factors of 18 are 3 and 2** because 18 is the **product of** **3² ****×**** 2**.

**Example**: We know that **7 and 3 are factors of 21** but they're also **prime factors**

of 21, since both **7 and 3 are prime** numbers.

5) The numbers **0 and 1 are neither prime nor composite**.

Now get a pencil, an eraser and a note book, copy the questions,

do the **practice** exercise(s), then **check your work** with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice Exercise 1:**

1) Express these numbers as the product of 2 factors. Make one of them a prime number:

a) 25 | b) 93 | c) 50 | d) 24 | e) 34 | f) 63 |

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**2) **Express the numbers above as the product of **only prime factors**. (use your answer from #1).

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**( solutions )**

Now continue with the lesson.

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**Greatest Common Factor (GCF)**

The ** Greatest Common Factor** (GCF) of a set of

since it is

**All the numbers** in the set **can be divided by** or

**are divisible by** the **Greatest Common Factor**.

** Example:** Because

2, 3, and 6 are common factors of 6 and 12 .

We name "6" -- the

will divide into both 6 and 12 to give a whole number (integer) quotient

**Example:** 2 is a common factor of 4 and 8 but 4 is the **Greatest Common** Factor.

**Hint:
**Since we're looking for the

So if we're factoring 26 and 39 to find their

26 = 13 × 2 and 39 = 13 × 3, so GCF = 13.

Now get a pencil, an eraser and a note book, copy the questions,

do the **practice** exercise(s), then **check your work** with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice Exercise 2:**

1) Factor these numbers to find their Greatest Common Factor.

a) 25 and 30 | b) 33 and 22 | c) 150 and 75hint: think money |
d) 24 and 32 | e) 63 and 18 |

**( solutions )
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Now continue with the lesson.

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**Lowest or Least Common Multiple (LCM)**

A **multiple** is the **product** of a given **number** and any other **whole number** or **integer**.

**Example:** 9, 21 and 30 are all multiples of 3.

And, 10, 50 and 60 are all multiples of 10 and of 5 and of 2.

The** Lowest or Least Common Multiple** (LCM) of a set of numbers is the **smallest** number that is a ** multiple of each and every number in the set**.

To find **the LCM of** the **numbers** in the set, we

**divide their product** **by their Greatest Common Factor**.

The **LCM of** a set of **prime numbers** is the **product of the primes**.

**Example:** **12** is the **LCM of 6 and 12**, because 6 is their GCF and .

Similarly, **24** is the **LCM of 6 and 8** , because 2 is their GCF and .

And **15** is the **LCM of 3 and 5** , because both **3 **and** 5 are prime.**

But instead of using the formula to find the LCM of two composite numbers, let's look at another example where we factor the numbers to understand why the LCM is the product of the numbers divided by the Lowest Common Factor.

**Example:** We want to find the Lowest Common Multiple of 6 and 4.

We know that** ****6 = 2 × 3** and that **4 = 2 × 2**, so the number **2 × 2** **× 3** includes both

the 4 (2 × 2) and the 6 (2 × 3), therefore it is the ** LOWEST** COMMON MULTIPLE.

What we're really saying is that

both 6 and 4 to give a whole number quotient with no remainder.

12 ÷ 6 = 2 and 12 ÷ 4 = 3

So, 12 is a multiple of 6, and 12 is a multiple of 4.

It is the **smallest or least multiple** of both 6 and 4 or the LCM.

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Now get a pencil, an eraser and a note book, copy the questions,

do the **practice** exercise(s), then **check your work** with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice Exercise 3:**

1) Use the formula (LCM = product ÷ LCF) to find the Lowest Common Multiple for:

a) 4 and 5 | b) 6 and 8 | c) 9 and 21 | d) 12 and 15 | e) 60 and 10 |

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2) Factor these numbers then find their LCM.

a) 4 and 5 | b) 4 and 12 | c) 9 and 15 | d) 21 and 14 | e) 6 and 10 |

**( solutions )**

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**Solutions**

1) There are many different solutions. Just check that the product is correct.

a) 25 = 5 × 5 |
b) 93 = 3 × 31 |
c) 50 = 5 × 10 |
d) 24 = 8 × 3 |
e) 34 = 2 × 17 |
f) 63 = 3 × 21 |

**2) **Express the numbers above as the product of **only prime factors**.

a) 25 = 5 × 5 |
b) 93 = 3 × 31 |
c) 50 = 5 × 5 × 2 = 5² × 2 |

d) 24 = 2 × 2 × 2 × 3 = 2³ × 3 | e) 34 = 2 × 17 | f) 63 = 3 × 3 × 7 = 3²× 7 |

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1) Factor these numbers to find their Greatest Common Factor.

a) 25 = 5 × 5 30 = 5 × 6 GCF = 5 |
b) 33 = 3 × 11 22 = 2 × 11 GCF = 11 |
c) 150 = 75 × 6 75 = 75 × 1 GCF = 75 |
d) 24 = 8 × 3 32 = 8 × 4 GCF = 8 |
e) 63 = 9 × 7 18 = 9 × 2 GCF = 9 |

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1) Use the formula (LCM = product ÷ LCF) to find the Lowest Common Multiple for:

a) 4 × 5 ÷ 1 = 20 |
b) 6 × 8 ÷ 2 = 24 |
c) 9 × 21 ÷ 3 = 63 |
d) 12 × 15 ÷ 3 = 60 |
e) 60 × 10 ÷ 10 = 60 |

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2) Factor these numbers then find their LCM.

a) 4 = 4 × 1 5 = 5 × 1 LCM = 20 |
b) 4 = 2 × 2 12 = 2 × 2 × 3 LCM = 12 |
c) 9 = 3 × 3 15 = 5 × 3 LCM = 45 |
d) 24 = 2³ × 3 32 = 2 ^{ 5} soLCM = 2^{ 5} × 3 = 96 |
e) 6 = 2 × 3 18 = 3² × 2 LCM = 18 |

*(all content of the MathRoom Lessons **© Tammy the Tutor; 2004 - ).*