In the second post of this series, we looked at how primes are used in cyber security. In this sub-series, we're going to have look at a number of important properties of prime numbers and look at why these must be true. I'll warn you now that some of the proofs are quite complex and require quite a wide understanding of algebra and number theory so it's not a big deal if they're confusing. But make sure to at least pay attention to the facts themselves as they are often useful at some point, and if not they are still interesting. For this first fact we'll be looking at Prime Factors.
1. All composite numbers can be expressed as the product of two or more prime factors.
For example, 63 is the product of 7x9, but then you can divide again to make it 63 = 3x3x7, or 3^2 x7. This is true for all composite numbers, but not primes, as they only have one prime factor, being themselves. This idea of prime factorisation is used for various different reasons, including calculating the total number of factors a number has. To do this, you factorise the number, for example 120 = 2^3 x 3 x 7, or showing full exponents,
120 = 2^3 x 3^1 x 5^1. You then add one to each of the exponents (in bold), and multiply them together, for example
(3+1)(1+1)(1+1) = 4x2x2 = 2^4 = 16. Therefore, 120 has 16 factors.
Look out for more of these prime posts.
1. All composite numbers can be expressed as the product of two or more prime factors.
For example, 63 is the product of 7x9, but then you can divide again to make it 63 = 3x3x7, or 3^2 x7. This is true for all composite numbers, but not primes, as they only have one prime factor, being themselves. This idea of prime factorisation is used for various different reasons, including calculating the total number of factors a number has. To do this, you factorise the number, for example 120 = 2^3 x 3 x 7, or showing full exponents,
120 = 2^3 x 3^1 x 5^1. You then add one to each of the exponents (in bold), and multiply them together, for example
(3+1)(1+1)(1+1) = 4x2x2 = 2^4 = 16. Therefore, 120 has 16 factors.
Look out for more of these prime posts.
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