LCD, GCF, Prime Factorization, Composite Number, Prime Number - definitions and meanings

This is a reference guide to LCD, GCF, Prime Factorization, Composite Number with meanings and definitions for basic understanding and as a reasearch tool for a student and beginners.

What is LCD least common denominator, GCF -greatest common factor, Mathematics, math, Prime Factorization, Prime Number, composite number, meanings, description, definition.

LCD

Definition of Least Common Denominator (LCD)
  • Least Common Denominator is the least common multiple of two or more of fractions.
More about Least Common Denominator (LCD)
  • It is also called as lowest common denominator.
  • Steps to find the Least Common Denominator:
1. First find the multiples of the denominators.

2. Identify the least common multiple.
Example of Least Common Denominator (LCD)
  • To find the least common denominator of the fractions and , first we need to find the multiples of the denominator 7 i.e. 7, 14, 21, 28, 35 . . . and the multiples of the denominator 3 i.e. 3, 6, 9, 12, 15, 18, 21, 24 . . . The least common multiple of the denominators 7 and 3 is 21.
So, the least common denominator of the fractionsand is 21.
Solved Example on Least Common Denominator (LCD)
Find the least common denominator of the given expression.
+
Choices:
A. 12
B. 16
C. 4
D. 21
Correct Answer: A
The least common denominator of two or more non-zero denominators is actually the smallest whole number that is divisible by each of the denominators. There are two widely used methods for finding the least common denominator.
Actually, this is the same basic idea behind finding the Least Common Multiple (LCM) for whole numbers (without the fractional parts).
Note: In the examples below, we'll be adding three fractions instead of the usual two because the principles are the same. This will give you a better understanding of the process. And in the "Pulling Everything Together" section, we will be adding four fractions.
Let's take a look at...
Method 1:
To find the least common denominator, simply list the multiples of each denominator (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.
Example: Suppose we wanted to add 1/5 + 1/6 + 1/15. We would find the least common denominator as follows...
·         First we list the multiples of each denominator.
Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,...
Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,...
Multiples of 15 are 30, 45, 60, 75, 90,....
·         Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.
·         Therefore, the least common denominator of 1/5, 1/6 and 1/15 is 30.
This method works pretty good. But, adding fractions with larger numbers in the denominators it can get pretty messy.
So hold that thought for a moment, as we look at another way to find a least common denominator for adding these same fractions.

Method 2:
To find the least common denominator using this method, factor each of the denominators into primes. Then for each different prime number in all of the factorizations, do the following...
1.        Count the number of times each prime number appears in each of the factorizations.
2.        For each prime number, take the largest of these counts.
3.        Write down that prime number as many times as you counted for it in step #2.
4.        The least common denominator is the product of all the prime numbers written down.
Example: We'll use the same fractions as above: 1/5, 1/6 and 1/15.
·         Factor into primes (Click here to see our table of prime numbers.)
o        Prime factorization of 5 is 5 (5is a prime number)
o        Prime factorization of 6 is 2 x 3
o        Prime factorization of 15 is 3 x 5
Notice that the different primes are 2, 3 and 5.
·         Now, we do Step #1 - Count the number of times each prime number appears in each of the factorizations...
o        The count of primes in 5 is one 5
o        The count of primes in 6 is one 2 and one 3
o        The count of primes in 15 is one 3 and one 5

·         Step #2 - For each prime number, take the largest of these counts. So we have...
o        The largest count of 2s is one
o        The largest count of 3s is one
o        The largest count of 5s is one

·         Step #3 - Since we now know the count of each prime number, you simply  - write down that prime number as many times as you counted for it in step #2.
Here are the numbers...
2, 3, 5
·         Step #4 - The least common denominator is the product of all the prime numbers written down.
2 x 3 x 5 = 30
·         Therefore, the least common denominator of 1/5, 1/6 and 1/15 is 30.
As you can see, both methods end up with the same results.



 

Greatest common factor


Definition:


The greatest common factor (GCF) is the largest factor of two numbers. An understanding of factor is important in order to understand the meaning of GCF.

What are factors?

When two or more numbers are multiplied in a multiplication problem, each number is a factor in the multiplication.

Take a look at the following multiplication problem:

2 times 8 times 3.

2 is a factor. 8 is also a factor.

You can find all factors of a number by finding all numbers that divide the number.

Find all factors of 36:

Start with 1. 1 divide 36, so 1 is a factor.

2 divides 36, so 2 is factor

3 divides 36, so 3 is a factor.

If you continue with this pattern, you will find that 1,2,3,4,6,9,12,18,and 36 are all factors of 36.

An easier way to handle the same problem is to do the following:

1 times 36 =36

2 times 18 = 36

3 times 12 =36

4 times 9=36

6 times 6 =36

9 times 4 =36.

Note:When the factors start to repeat, you have found them all.

In our example above, the factors started to repeat at 9 times 4 =36 because you already has 4 times 9 =36.

Therefore, we have found them all.

Now that you have understood how to get the factors of a number, it is going to be straightforward to to get the greatest common factor.

Whenever you are talking about greatest common factor, you are referring to 2 or 3 numbers. Here, we will concern ourselves with just 2

The GCF of two numbers is the largest factor of the two numbers.

For instance, find GCF of 16 and 24 written as GCF(16,24).

The factors for 16 are 1, 2, 4, 8,and 16

The factors for 24 are 1,2, 3,4,6,8,12, and 24.

The largest factor for both numbers have in common is 8, so GCF(16,24) = 8.

Find GCF(7,12)

The factors for 7 are 1 and 7.

The factors for 12 are 1, 2, 3, 4,6,and 12

The largest number both factors have in common is 1, SO GCF(7,12)=1

 

Prime Numbers

A natural number is called a prime number (or a prime) if it is bigger than one and has no divisors other than 1 and itself. For example, 5 is prime, since no number except 1 and 5 divides it. On the other hand, 6 is not a prime (it is composite), since 6 = 2 · 3. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any positive integer n can be expressed as the product of powers of primes in a way that is unique except for a possible reordering of the factors. This theorem requires excluding 1 as a prime. There are infinitely many primes, as demonstrated by Euclid around 300 BC.

Other helpful hints:
  • Zero and 1 are not considered prime.
  • The only even prime number is 2. All other even numbers can be divided by 2. And remember, if it ends in ZERO it's also even.
  • No number greater than the prime number 5 ends in a 5. Because, if it ends in a 5 can be divided by 5.
  • Any number where the "sum" of the digits add up to a number that is divisible by "3" is not prime.
Example: The number 1,269 is not prime because the sum of the digits (1 + 2 + 6 + 9 = 18) and since "18" is evenly devisable by 3, the number 1,269 is NOT a prime number.
  • Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime.
To test whether a number is a prime number, try dividing it by 3, then by 5, then by 7, and so on.
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67 71
73 79 83 89 97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349



Composite number

A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number.
So, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a × b, then n is composite. By definition, every integer greater than one is either a prime number or a composite number. The number one is a unit – it is neither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself.
The first 105 composite numbers (sequence A002808 in OEIS) are
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140.
Every composite number can be written as the product of two or more (not necessarily distinct) primes; furthermore, this representation is unique up to the order of the factors.
A composite number nis a positive integer n>1which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called "composites" for short) are 4, 6, 8, 9, 10, 12, 14, 15, 16, ... (Sloane's A002808), whose prime decompositions are summarized in the following table. Note that the number 1 is a special case which is considered to be neither composite nor prime.
n
n
4
2^2
20
2^25
6
2·3
21
3·7
8
2^3
22
2·11
9
3^2
24
2^3·3
10
2·5
25
5^2
12
2^23
26
2·13
14
2·7
27
3^3
15
3·5
28
2^27
16
2^4
30
2·3·5
18
2·3^2
32
2^5
The nth composite number can be generated using the Mathematica code
  Composite[n_Integer] :=
    FixedPoint[n + PrimePi[#] + 1&, n]
There are an infinite number of composite numbers.
The composite number problem asks if there exist positive integers mand nsuch that N=mn.
A composite number Ccan always be written as a product in at least two ways (since 1·Cis always possible). Call these two products
 C=ab=cd,
(1)
then it is obviously the case that c|ab(c divides ab). Set
 c=mn,
(2)
where mis the part of cwhich divides a, and nis the part of cwhich divides b. Then there are pand qsuch that
a
=
mp
(3)
b
=
nq.
(4)
Solving ab=cdfor dgives
 d=(ab)/c=((mp)(nq))/(mn)=pq.
(5)
It then follows that
S
=
a^2+b^2+c^2+d^2
(6)
=
m^2p^2+n^2q^2+m^2n^2+p^2q^2
(7)
=
(m^2+q^2)(n^2+p^2).
(8)
It therefore follows that a^2+b^2+c^2+d^2is never prime! In fact, the more general result that
 S=a^k+b^k+c^k+d^k
(9)
is never prime for kan integer >=0also holds (Honsberger 1991).

Prime Numbers and Composite Numbers

A Prime Number can be divided evenly only by 1 or itself.
And it must be a whole number greater than 1.
  • When a number can be divided up evenly it is a Composite Number
  • When a number can not be divided up evenly it is a Prime Number

Prime Factorization – ladderize prime factor

"Prime Factorization" is finding which prime numbers multiply together to make the original number.
A Prime Number can be divided evenly only by 1 or itself.
And it must be a whole number greater than 1.

Example 1: What are the prime factors of 12 ?

It is best to start working from the smallest prime number, which is 2, so let's check:
12 ÷ 2 = 6
Yes, it divided evenly by 2. We have taken the first step!
But 6 is not a prime number, so we need to go further. Let's try 2 again:
6 ÷ 2 = 3
Yes, that worked also. And 3 is a prime number, so we have the answer:
12 = 2 × 2 × 3

As you can see, every factor is a prime number, so the answer must be right.

Note: 12 = 2 × 2 × 3 can also be written using exponents as 12 = 22 × 3

We hope this research guide improves your learning about this subject.



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