I have my degree in mathematics and have tutored math for 7 years and oftentimes see confusion among students when it comes to prime numbers. The simple reason is the definition of prime numbers that most textbooks use. Most say that a prime number is any 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 any number that has only 2 factors, which generally IS 1 and itself, with the exception of the number 1. That only has 1 factor, because the only numbers that multiply to give you 1 is 1 and 1. All other 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, you might say which make the explanation a lot clearer.
*First 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 root that is a whole number is not prime. By applying these rules you can eliminate
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, 54,55,56,57,58,60,62,63,64,65,66,68,69,70,72,74,75,7 6,77,78,80,81,82,84,85,86,88,90,92,93,94,95,96,98,99 , and 100 as possible prime numbers.
This leaves only the numbers 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. All of these numbers 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 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.
So the rules I give were enough to determine all the 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 learn how to find the prime numbers between 1 and 100.