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.
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.
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.
What is the difference between Prime Number and Prime Factors?
¤ 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.