## Saturday, May 3, 2014

During my nearly thirteen years of tutoring math, there has been confusion among students when it comes to prime numbers. Part of the reason is the definition of prime numbers seen in most textbooks, which is that a prime number is any natural number whose only factors are 1 and itself. Using that definition, almost everyone will say that 1 is a prime number, when in fact it is not.

The true definition is that a prime number is any natural number that has only two factors, which generally is 1 and itself. That definition excludes 1, which only has one factor, because the only numbers that multiply to give you 1 is 1 and 1. All other natural numbers that are not prime are called composite numbers.
I have a way of teaching students the prime numbers from 1 to 100 using a few rules and tricks which make the explanation a lot clearer. You can start eliminating numbers as possible prime numbers based on the following:

*Any number that ends in 0 is divisible by 10 and 5.
*Any number ending in 5 is divisible by 5.
*Any even number is divisible by 2.
*Any number whose digits added is divisible by 3.
The entire number is divisible by 3. An example of this is 39. 3+9 = 12, which is divisible by 3, so 39 is also divisible by 3.
*Any 2 digit number where both digits are the same is divisible by 11.
*Also any number that has a square that is a whole number is not prime.

By applying these rules you can eliminate the following as possible prime numbers:
4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,
50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,74,75,76,77,78,80,81,82,84,85,86,88,90,
92,93,94,95,96,98,99, and 100.

The numbers remaining (2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,87,89,91,97) are prime.

What if one wants to know the prime numbers greater than 100? There are a few good methods to read and study. The Sieve of Eratosthenes is an ancient method to find any prime number up to a specified number. For a faster, although more complex method to find prime numbers up to a specified number, one can use the Sieve of Atkin. The Sieve of Atkin is an optimized version of the Sieve of Eratosthenes and involves dividing numbers by 60, getting the remainder, and using the remainder to solve more complex quadratic equations. Most of the time a computer or calculator would be needed to solve such equations.

The rules I give are enough to determine prime numbers between 1 and 100, which is generally enough for all practical purposes. I hope this information will be useful for any student to find the prime numbers between 1 and 100.