December 9th, 2014, 03:20 PM  #1 
Newbie Joined: Dec 2014 From: alexanderia Posts: 1 Thanks: 0  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. 
December 9th, 2014, 04:18 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
shaimaa saif, what's your relationship to Nitin Gupta?

December 9th, 2014, 06:01 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,060 Thanks: 1396 
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). 
December 9th, 2014, 06:24 PM  #4 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 371 Thanks: 68 
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. 
December 9th, 2014, 06:41 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,973 Thanks: 2296 Math Focus: Mainly analysis and algebra 
A more interesting challenge might be to show that composite numbers have one and only one decomposition into a product of prime numbers.

December 9th, 2014, 06:48 PM  #6  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,596 Thanks: 620 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
December 10th, 2014, 07:13 AM  #7 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  
November 11th, 2015, 12:17 PM  #8 
Newbie Joined: Nov 2015 From: Australia Posts: 1 Thanks: 0 
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.

November 16th, 2015, 02:55 PM  #9  
Senior Member Joined: Oct 2013 From: New York, USA Posts: 561 Thanks: 79  Quote:
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.  
November 17th, 2015, 08:00 AM  #10  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,734 Thanks: 707  Quote:
 

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