A Guide to the GRE/Prime Numbers

Prime Numbers

A prime number is a number only divisible by itself and 1.

Prime numbers between 1 and 100 are as follows:

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

Some numbers like 51, 57 or 93 may look like prime numbers but are in fact divisible by an oddball prime number. These three are divisible by 17, 19 and 31, respectively.

Non-prime numbers can be divided into their prime factors.

A “factor” is a number by which a larger number can be divided. Non-prime numbers can be broken down into their prime factors using a factor tree.

24

         |     |
       2       12
              |     |
            2        6
                    |   |
                   2     3

 24 = 2(2)(2)(3) or 23(3)

This expression is known as the “prime factorization” of 24. When factoring numbers, it is typically easiest to divide by 2's and 3's when possible.

Practice

1. What is the prime factorization of 100?

2. How many prime numbers are greater than 50 but less than 60?

3. Express 372 as a product of prime numbers.

Answers to Practice Questions

1. 2²(5²) or (2)(2)(5)(5)

Prime factorization can be achieved by drawing a factor tree - picking any two numbers that a number divides into, and dividing downward.

100

          /      \
        4        25
      /    \     /    \
    2      2  5     5

The bottoms of the branches are the prime factors. It doesn't matter which two numbers are picked first. All factor trees lead to the same numbers.

100

          /      \
        5        20
      	      /    \

2 10 / \

	          2      5

2. Two - 53 and 59.

51 is divisible by 17, while 57 is divisible by 19.

3. 2²(3)(31) or 2(2)(3)(31)

            372
         /       \
       2        186
                /     \
              2       93
                      /    \
                    3     31

As you can see, it is usually easier when factoring to take out 2's or 3's at first.