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December 9th, 2014, 03:20 PM   #1
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prime numbers and composite numbers

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.

Example: 7 can only be divided evenly by 1 or 7, so it is a prime number.

But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it is a composite number).

Let me explain ...

Some whole numbers can be divided up evenly, and some can't!

Example:

6 can be divided evenly by 2, or by 3:

6 = 2 × 3

Like this:

6 can be divided into 2 groups of 3 or into 3 groups of 2.

But 7 cannot be divided up evenly:

7 is Prime.

And we give them names:

When a number can be divided up evenly, it is a Composite Number.
When a number cannot be divided up evenly, it is a Prime Number.
So 6 is Composite, but 7 is Prime.


And that explains it ... but there are some more details ...



Not Into Fractions

We are only dealing with whole numbers here! We are not going to cut things into halves or quarters.

Not Into Groups of 1

OK, we could have divided 7 into seven 1s (or one 7) like this:

7 is Prime
7 = 1 x 7
But we could do that for any whole number!

So we should also say we are not interested in dividing by 1, or by the number itself.

It is a Prime Number when it can't be divided evenly by any number
(except 1 or itself).


Example: is 7 a Prime Number or Composite Number?

7 is Prime

We cannot divide 7 evenly by 2 (we get 2 lots of 3, with one left over).
We cannot divide 7 evenly by 3 (we get 3 lots of 2, with one left over).
We cannot divide 7 evenly by 4, or 5, or 6.
We can only divide 7 into one group of 7 (or seven groups of 1):

7 is Prime
7 = 1 x 7


So 7 can only be divided evenly by 1 or itself:

So 7 is a Prime Number

And also:
It is a Composite Number when it can be divided evenly
by numbers other than 1 or itself.

Like this:

Example: is 6 a Prime Number or Composite Number?

6 can be divided evenly by 2, or by 3, as well as by 1 or 6:

6 = 1 × 6
6 = 2 × 3

So 6 is a Composite Number

Sometimes a number can be divided evenly many ways:

Example: 12 can be divided evenly by 1, 2, 3, 4, 6 and 12:

1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

So 12 is a Composite Number

And note this:

Any whole number greater than 1 is either Prime or Composite

What About 1?

Years ago 1 was included as a Prime, but now it is not:

1 is not Prime and also not Composite.

Factors

We can also define a Prime Number using factors.


"Factors" are numbers we multiply
together to get another number.

And we have:

When the only two factors of a number are 1 and the number,
then it is a Prime Number

It means the same as our previous definition, just stated using factors.

And remember this is only about Whole Numbers (1, 2, 3, ... etc.), not fractions or negative numbers. So don't say "I could multiply ½ times 6 to get 3" OK?

Examples:

3 = 1 × 3
(the only factors are 1 and 3) Prime

6 = 1 × 6 or 6 = 2 × 3
(the factors are 1,2,3 and 6) Composite
Examples From 1 to 14

Factors other than 1 or the number itself are highlighted:

Number
Can be Evenly
Divided By
Prime, or
Composite?
1
(1 is not considered prime or composite)
2
1, 2
Prime
3
1, 3
Prime
4
1, 2, 4
Composite
5
1, 5
Prime
6
1, 2, 3, 6
Composite
7
1, 7
Prime
8
1, 2, 4, 8
Composite
9
1, 3, 9
Composite
10
1, 2, 5, 10
Composite
11
1, 11
Prime
12
1, 2, 3, 4, 6, 12
Composite
13
1, 13
Prime
14
1, 2, 7, 14
Composite
...
...
...
So when there are more factors than 1 or the number itself, the number is Composite.

A question for you: is 15 Prime or Composite?

Why All the Fuss about Prime and Composite?

Because we can "break apart" Composite Numbers into Prime Number factors.

2 and 2 and 3
It is like the Prime Numbers are the basic building blocks of all numbers.

And the Composite Numbers are made up of Prime Numbers multiplied together.

Here we see it in action:



2 is Prime, 3 is Prime, 4 is Composite (=2×2), 5 is Prime, and so on...

Example: 12 is made by multiplying the prime numbers 2, 2 and 3 together.

12 = 2 × 2 × 3

The number 2 was repeated, which is OK.

In fact we can write it like this using the exponent of 2:

12 = 22 × 3


And that is why they are called "Composite" Numbers because composite means "something made by combining things".

Last edited by skipjack; December 9th, 2014 at 03:42 PM.
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December 9th, 2014, 04:18 PM   #2
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shaimaa saif, what's your relationship to Nitin Gupta?
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December 9th, 2014, 06:01 PM   #3
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Definition: if p, q and n are whole numbers greater than zero, and pq = n, we say that p and q are divisors of n.

This means that 1 has just one divisor (namely 1), 2 has two divisors (1 and 2), 3 has two divisors (1 and 3), 4 has three divisors (1, 2 and 4), and so on.

If n has exactly two divisors, it is called a prime number (or a prime).
If n has more than two divisors, it is called composite (or a composite number).
Note that 1 is neither prime nor composite.

If n is composite, it can be expressed as a product of primes. For example, 12 = 2 × 2 × 3.
It can be shown that such a prime factorization of a composite number is unique (apart from the order in which the primes appear).
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December 9th, 2014, 06:24 PM   #4
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I'm not sure, shaimaa, why you've given this lecture on prime and composite numbers. I doubt that anyone who posts to this forum needs it.

Perhaps a more interesting challenge would be to prove that there are an infinite number of primes.
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December 9th, 2014, 06:41 PM   #5
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A more interesting challenge might be to show that composite numbers have one and only one decomposition into a product of prime numbers.
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December 9th, 2014, 06:48 PM   #6
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Quote:
Originally Posted by Timios View Post
I'm not sure, shaimaa, why you've given this lecture on prime and composite numbers. I doubt that anyone who posts to this forum needs it.

Perhaps a more interesting challenge would be to prove that there are an infinite number of primes.
Well, the thread is in the Pre-Algebra Forum, so I think it's appropriate.

-Dan
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December 10th, 2014, 07:13 AM   #7
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Quote:
Originally Posted by topsquark View Post
Well, the thread is in the Pre-Algebra Forum, so I think it's appropriate.
Maybe, if the content wasn't lifted from Nitin Gupta.
Thanks from topsquark
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November 11th, 2015, 12:17 PM   #8
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Some interesting facts and puzzles about Prime Numbers and Magic Squares, Smith Numbers, and Arithmetic and Palindromic Primes on this blog: Glenn Westmore -Glenn Westmore.
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November 16th, 2015, 02:55 PM   #9
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Quote:
Originally Posted by v8archie View Post
A more interesting challenge might be to show that composite numbers have one and only one decomposition into a product of prime numbers.
Say that x and y are prime factors of a composite number c. In order for there to be multiple prime factorizations of c, one of these would need to be true, and neither is true:

1. xy is prime, which it isn't because x and y are factors of it.

2. xy = ab, where a, b, x, and y are all prime. This is impossible. Therefore what we want to prove is that if x and y are prime, a composite number xy cannot be a multiple of any prime numbers other than x and y.
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November 17th, 2015, 08:00 AM   #10
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Quote:
Originally Posted by EvanJ View Post
Say that x and y are prime factors of a composite number c. In order for there to be multiple prime factorizations of c, one of these would need to be true, and neither is true:

1. xy is prime, which it isn't because x and y are factors of it.

2. xy = ab, where a, b, x, and y are all prime. This is impossible. Therefore what we want to prove is that if x and y are prime, a composite number xy cannot be a multiple of any prime numbers other than x and y.
Yes! How do you do that?
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