In the Prime of Its Life

The first primes less than 100  and rules to easily find

Here is a common scenario:

You are taking the SAT test and you are asked how many prime numbers there are between 30 and 80.  You know what prime numbers are and you know what prime numbers are but you have limited time and no table to help you.  You decide to skip the question because you do not have enough time to finish it by the end of the test.

This is a very common test question yet it is not trivial because most people have not memorized the prime numbers so here is a quick was to determine the prime numbers. 

1   2   3   4   5   6   7   8   9   10            (4 primes)

11 12 13 14 15 16 17 18 19 20            (4 primes)

21 22 23 24 25 26 27 28 29 30            (2 primes)

31 32 33 34 35 36 37 38 39 40            (2 primes)

41 42 43 44 45 46 47 48 49 50            (3 primes)

51 52 53 54 55 56 57 58 59 60            (2 primes)

61 62 63 64 65 66 67 68 69 70            (2 primes)

71 72 73 74 75 76 77 78 79 80            (3 primes)

81 82 83 84 85 86 87 88 89 90            (2 primes)         

91 92 93 94 95 96 97 98 99 100          (1 prime)

                                                 
 The sieve of Eratosthenesis is a methodology for finding all prime numbers.  Without going into how the sieve of Eratosthenes works, I will say the methodology I present will be similar but not the same and it will be easier to use for numbers between 1 and 100.  

After 2, all even numbers are not prime.  
After 5, all numbers that end in 5 are not prime.
No numbers that end in 0 are prime.

Now here is the trick.  If you are looking at the columns where the numbers end in a 1, 3, 7, or 9, you will see there are quite a few primes.  Of these numbers that end in 1, 3, 7, and 9, all numbers except 1, 3, 49, 77, 91, are divisible by 3.

Okay lets check this out.
21÷3=21              33÷3=11               27÷3=9               39÷3=13
51÷3=14              63÷3=21              57÷3=19              81÷3=27
81÷3=27              93÷3=31              87÷3=29              69÷3=23 

You can easily find any prime on numbers less than 100.  If you are asked if 87 is prime, you will say no it isn't because it is divisible by 3.  If you use the 3 divisibility rule, it is even easier.  

Also one more trick for those taking the SAT or any standardized test, the number of primes in each row after the tens are:
 2, 2, 3, 2, 2, 3, 2, 2, 1 and the ones and 10's each have 4.  Refer to the table above.
If you are asked the question, "How many prime numbers are there between 30 and 80?"  You can answer 16.  2+3+2+2+3+2+2=16


                                                              

3 Times is a charm

3 divisibility rule

If a number  is divisible by 3, then the sum of the digits of the number are also divisible by 3.

As an example:

63, 123, 2223 are all divisible by 3.  We can easily determine this in our heads without a calculator by adding the digits and then dividing by 3 as follows:

6+3=9 

9÷3=3

63 is evenly divisible by 3.

If we use a calculator, we find 63÷3=21

1+2+3=6

6÷3=2

123 is divisible by 3.

If we use our a calculator, we find 123÷3=41

2+2+2+3=9

9÷3=3

2223 is divisible by 3
If we use our calculator, we find 2223÷3=741

89, 173, and 3113 are not divisible by 3.  Here is why:

8+9=17 and 17 is not divisible by 3

1+7+3=11 and 11 is not divisible by 3

3+1+1+3=8 and 8 is not divisible by 3

This is a very simple rule yet very powerful.  There are other rules that I will present a little later but this is one of the most important rules and this will help you through many tests like the SAT and ACT.

Try these on your own:

Are these numbers divisible by 3?  Add the digits and divide.  Check with your calculator.

1234, 174, 12761, 25437

Hint all are except 12761.

Prime Time 

What are prime numbers?

A prime number is a positive integer that can only be divided evenly by by 2 different integers 1 and itself.  5 can only be divided evenly by 5 and 1 so this number is prime.  6 can be divided evenly by 2, 3, 6, and 1 so 6 would not be a prime number.

Some fun facts:

• 1 is not a prime number.  It is defined as not prime and is not divisible by 2 different integers.  It is only divisible by 1.
• 2 is the only even prime number.
• Numbers that are greater than one and not prime are called composite numbers.
• There are infinitely many prime numbers.
• 0 is not a prime number.

How do we determine if a number is prime?

You can easily figure out if a number is prime by dividing by smaller prime numbers to see if it can be evenly divided.  If I want to find out if 11 prime, I can try to divide it by smaller prime numbers like 2,3,5, or 7.  11 cannot be evenly divided by any smaller prime factors therefore it is prime.  If I wanted to find out if 33 is prime, I can divide 33 by 3 and I find it will evenly divide into 33 with factors of 1,3,11, and 33, therefore it is not prime.

Following are the prime numbers less than 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Practice:

Find out if the following numbers are prime:  101, 123 149, and 171.

Definitions:

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.