Prime-al Instinct

How to know if a number is prime.

Knowing if a number is prime  may not exactly be an instinct.   However, it is actually fairly easy to determine if the number is prime.  Many schools teach students to just memorize the prime numbers under 100.  I have observed students being asked on a test if a number like 443 is prime.  Most students aren't able to answer that question.

There are many advanced topics on prime numbers and they are not easy to read and are hard to understand so we will avoid those concepts and teach an easy to understand concept.   We will start with the definition of  prime number which is a number whose factors are only 1 and itself.  So to be a non-prime or composite number, the number will havoc to be divisible by some number.  Because every number can be factored down to their prime numbers, we will divide by prime numbers and see if the number is prime.

We remember divisibility rules and know that a number that ends in a 3 cannot be divided by 2 or 5. We will start with:

443÷3 is not evenly divisible
443÷7 is not evenly divisible

443÷11 is not evenly divisible

443÷13 is not evenly divisible

443÷17 is not evenly divisible

443÷19 is not evenly divisible

443÷23 is not evenly divisible

The number is prime.  You might wonder why I stopped at 23 and did not test all the numbers up to 443.  The reason is that 443 is less than 23².  23²=529.    If the number is composite, then it must consist of two different factors and if the number  has a prime factor higher that 23,  there must be a corresponding  prime factor lower than 23.  This is because 23 times any number larger than 23 would be more than 443.  Therefore if 443 is not prime, there will be a number less than 23 which is its factor and we already tested all prime factors less than 23.

Exercise

Try and see if the following numbers are prime:  299, 137, 441, and 537.

Divide evenly:  To divide evenly means that if you divide one number by another there is no remainder.  We can make this easier by saying that it will not produce a non integer.  4 divided by 3 = 1.33333..... which is not an integer so 4 is not evenly divisible by 3.

Composite number:  A number that is composed of two different integers.  With the exception of 1 and 0, a composite number is an integer that is not a prime number.

Answers:  137 and 537 are prime.