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Arithmetic and its Fundamental Theorem

The natural number system-the set of numbers that comes to us naturally- is comprised of mainly composite and prime numbers. Numbers obtained by multiplying two other numbers are composite numbers while prime numbers are numbers whose factors are themselves and 1 only. Examples of composite numbers include numbers such as 6 which is obtained by multiplying 2 and 3. Prime numbers 2 and 3 are said to be the factors of 6 or they are divisors of 6. Prime numbers are so many and they are typically known as numbers with two factors only-itself and 1.

Examples of prime numbers include

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …

Prime numbers are infinitely many. You might have noticed that 1 was not included as a prime number. It is actually not a prime number because it does not satisfy the definition but it is a number in a category of its own.

The relationship between composite and prime numbers has been made clearer by the Fundamental Theorem of Arithmetic. The theorem states that every natural number is either prime or can be expressed uniquely as a product of prime numbers.

Examples are:

6 = 2 x 3

12 = 2 x 2 x 3

24 = 2 x 2 x 2 x 3

120 = 2 x 2 x 2 x 3 x 5     etc.

These natural numbers cannot be expressed as products of prime factors more than these unique ways. All other attempt comes back to these patterns so long the factors are prime numbers.

From the fact of this theorem one may consider prime numbers as ‘’blocks’’ that built the natural number system. In the branch of mathematics known as the number theory, Karl Gauss, a notable mathematician considered the Fundamental Theorem of Arithmetic as very important.

As an exercise, it is rather interesting to write natural numbers of up to 120 as products of their prime factors or simply as a prime and you will see the beauty of the Fundamental theorem of Arithmetic.

We set you out on the exercise as follows:

(Natural numbers  2, 3,5,7,11,13,17,19,23,29,31,…,119 are all prime numbers.)

1 2 3 4=2×2

5 6=2×3 7 8=2x2x2

9=3×3 10=2×5 11 12=2x2x4…

…                            …                                 …                                 …

…                            …                                 …                                 …

120=2x2x2x3x5

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