Just as atoms are the building blocks of elements, prime numbers are the building blocks of arithmetic. This comparison is justified when we consider that all composite numbers can be uniquely generated by a specific product of prime numbers. This last statement is one of the most important in number theory and is known as the Fundamental Theorem of Arithmetic.
If you have forgotten your definitions, a Prime number is any number that is divisible only by the number 1 and itself. The first prime numbers are 2, 3, 5, 7, 11, and 13; 2 being the only even prime. IN compound number is a number that has at least one other factor besides the number 1 and itself. It should be noted that for a number to be prime it must be odd, but not all odd numbers are prime. Tea The fundamental theorem of arithmetic states that every composite number can be uniquely written, regardless of the order of the factors, by a product of prime numbers.
Take the number 56 for example. We can break it down using a factor tree and arrive at the following prime number decomposition: 2 x 2 x 2 x 7. When factoring a composite number into prime factors, all you need to do is become familiar with the first 10 prime numbers. more or less. These are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. If a number is even, like 56, keep dividing it by 2 until you can’t; then proceed to eliminate the other prime factors accordingly. To clarify this procedure, take the composite number 212. Since it is even, divide it into
2 x 106. Since 106 is even, break it down into 2 x 53. Since 53 is prime, we write
212 = 2x2x53.
Historically, prime numbers were considered of interest for purely mathematical reasons. It wasn’t until the 1970s that prime number theory found itself intricately intertwined with RSA cryptography, or data scrambling. Prime numbers are now recognized as an integral part of the Internet commerce infrastructure by enabling secure transactions through computers. Without a solid understanding of these curious mathematical numbers, secure e-commerce would not be possible.
Tea The fundamental theorem of arithmetic It also provides us with an algorithm to find the lowest common denominator, gcf, of two numbers, used when adding or subtracting fractions. For example, suppose we have 15 and 39. We can break these numbers into primes as follows: 15 = 3 x 5 and 39 = 3 x 13. We can find the LCD taking the product of the distinct primes and using the common primes only once. Thus, since 3, 5, and 13 are distinct, and 3 is used twice, we find the LCD writing the product of these numbers to get LCD = 3x5x13 = 195.
So next time you think a number is a number, it’s a number. Remember that this is not the case at all. The wonderful world of prime numbers teaches us that no type of number should be taken for granted. Who knows? Perhaps if you think about it long enough, you could find some exotic application for some types of numbers that are now studied for purely mathematical reasons. Then you too could become famous.