** Prime Number vs Prime Factors **

The concept ‘factorization’ is defined on integers. Therefore, the factor of a number (integer) is another integer that can divide the original into a third integer without leaving a reminder. Factors for a number include 1 and the number itself. For an example, factors of 8 are 1,-1, 2, -2, 4, -4, 8 and -8.

**Prime Number**

A prime number is a natural number greater than one, which is divisible only by one and the number itself. Therefore, a prime has only two factors, one and the number itself. For an example, 5 is a prime number since it is divisible only by one and the number itself. Positive integers that have more than two factors are termed as composite numbers. Eight is a composite number since it has more than two factors. There is no formula to generate prime numbers. To establish a number as a prime, we have to demonstrate that it has no factors other than 1 and the number itself by using the mathematical method of division and potential factors.

**Prime Factors**

Every integer has at least two factors. Out of these factors, some can be prime numbers. These are called prime factors. In other words, a prime factor of a number is a factor of that number and also a prime number. Therefore 2 is a prime factor of 8. However, the other factors of 8 are not prime factors, 4 is not a prime factor of 8, because 4 is a composite number.

The procedure of expressing a whole number as a product of prime factors is called prime factorization. First, it will try to check for factors of 2 in the number, and remove as much as possible. Then try the next prime 3 and remove as many factors of 3 as possible. Repeat the process until the number is expressed as a product of prime numbers.

For an example, let us find the prime factors of 840.

*840 contains a factor of 2*

*840 = 2 ×420*

*420 contains a factor of 2*

*840 = 2 ×2×210*

*210 contains a factor of 2*

*840 = 2 ×2×2×105*

*105 has no prime factors of 2. Since 105 is divisible by 3, 3 is a prime factor of 105.*

*840 = 2 ×2×2×3×35*

*35 has no prime factors of either 2 or 3. But, since 35 is divisible by 5, 5 is a prime factor of 35.*

*840 = 2 ×2×2×3×5 ×7*

*7 is itself a prime number. Thus, 840 can be written as a product of prime factors as follows.*

*840 = 2 ×2×2×3× 5×7*

When we remove prime factors, the number on which we need to focus further attention is always getting smaller.

¤ A prime number has only two factors, one and the number itself. ¤ A prime factor of a number is a factor and also a prime number. |