| 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 factors of a number are the two or more numbers whose product equals the number.
2) A prime number is a whole number (integer) whose only factors are itself and one.
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 product of prime numbers.
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 |
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 two or more numbers, is the Greatest or largest number that is a factor of all the numbers in the set. We call it a common factor,
since it is common to all the factorizations of all the numbers in the set.
All the numbers in the set can be divided by or
are divisible by the Greatest Common Factor.
Example: Because 6 = 2 × 3, and 12 = 2 × 2 × 3 , we say that
2, 3, and 6 are common factors of 6 and 12 .
We name "6" -- the Greatest Common Factor -- since it is the greatest number that
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 Greatest Common Factor, we factor into big-as-we-can numbers.
So if we're factoring 26 and 39 to find their Greatest Common Factor, we'll start with 13 -- which is a pretty great number -- to find that
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 75 hint: think money |
d) 24 and 32 | e) 63 and 18 |
( solutions )
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 
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Similarly, 24 is the LCM of 6 and 8 , because 2 is their GCF and 
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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 12 is the smallest number that can be divided by
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 |
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 |
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 so LCM = 2 5 × 3 = 96 |
e) 6 = 2 × 3 18 = 3² × 2 LCM = 18 |
