A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
A natural number greater than 1 that is not prime is called a composite number.
For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers (2 × 3) that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
The prime numbers are the natural numbers greater than one that are not products of two smaller numbers.
Video: Prime number
Definition of a prime number
A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it. The numbers greater than 1 that are not prime are called composite numbers. In other words, is prime if items cannot be divided up into smaller equal-size groups of more than one item,or if it is not possible to arrange dots into a rectangular grid that is more than one dot wide and more than one dot high.For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder).
1 is not prime, as it is specifically excluded in the definition.
4 = 2 × 2 and 6 = 2 × 3 are both composite.
Demonstration, with Cuisenaire rods, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly
The divisors of a natural number are the numbers that divide evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea leads to a different but equivalent definition of the primes:
They are the numbers with exactly two positive divisors, 1 and the number itself.
What are the first 25 prime numbers?
The first 25 prime numbers (all the prime numbers less than 100) are:
No even number greater than 2 is prime because any such number can be expressed as the product .
Therefore, every prime number other than 2 is an odd number, and is called an odd prime.
Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9.
The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.
The set of all primes is sometimes denoted by (a boldface capital P) or by (a blackboard bold capital P).
What is prime factorization and prime factor?
Video: Prime factorization
Writing a number as a product of prime numbers is called a prime factorization of the number.
For example:
The terms in the product are called prime factors.
The same prime factor may occur more than once; this example has two copies of the prime factor . When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for instance, in the second way of writing the product above, denotes the square or second power of .
What is the importance of prime numbers?
The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic.
Fundamental theorem of arithmetic states that every integer larger than 1 can be written as a product of one or more primes.
More strongly, this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes, although their ordering may differ. So, although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers.
What are co-prime numbers?
This video discusses what a co-prime number is.
Finding prime numbers.
This video discusses how to find prime numbers.
Let's Review
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two ____ natural numbers.
A natural number greater than 1 that is not prime is called a ____ ____ .
1 is ____ prime.
Primes are the numbers with exactly two positive divisors, ____ and the ____ itself.
Every prime number other than 2 is an odd number, and is called an ____ ____ .
When written in the usual decimal system, all prime numbers larger than 5 end in ____ , ____ , ____ , or ____ .
Writing a number as a product of prime numbers is called a ____ ____ of the number.
____ states that every integer larger than 1 can be written as a product of one or more primes.
Answer
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smallernatural numbers.
A natural number greater than 1 that is not prime is called a composite number.
1 is not prime.
Primes are the numbers with exactly two positive divisors, 1and the numberitself.
Every prime number other than 2 is an odd number, and is called an odd prime.
When written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9.
Writing a number as a product of prime numbers is called a prime factorization of the number.
Fundamental theorem of arithmetic states that every integer larger than 1 can be written as a product of one or more primes.
is prime if 
. 
(a boldface capital P) or by 
(a blackboard bold capital P).
. When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for instance, in the second way of writing the product above, 
denotes the square or second power of 
