Chapter 15
There is a category of problems on the GRE that tests what could broadly be referred to as “Number Properties.” These questions are focused on a very important subset of numbers known as integers. Before we explore divisibility any further, it will be necessary to understand exactly what integers are and how they function.
Integers are whole numbers. That means that they are numbers that do not have any decimals or fractions attached. Some people think of them as counting numbers, that is, 1, 2, 3 … and so on. Integers can be positive, and they can also be negative. For instance, −1, −2, −3 … are all integers as well. And there’s one more important number that qualifies as an integer: 0.
So numbers such as 7, 15,003, −346, and 0 are all integers. Numbers such as 1.3, 3/4, and π are not integers.
Now let’s look at the rules for integers when dealing with the four basic operations: addition, subtraction, multiplication and division.
| integer + integer = always an integer | ex.: 4 + 11 = 15 |
| integer − integer = always an integer | ex. :−5 − 32 = −37 |
| integer × integer = always an integer | ex.: 14 × 3 = 42 |
None of these properties of integers turn out to be very interesting. But what happens when we divide an integer by another integer? Well, 18 ÷ 3 = 6, which is an integer, but 12 ÷ 8 = 1.5, which is not an integer.
If an integer divides another integer and the result, or quotient, is an integer, you would say that the first number is divisible by the second. So 18 is divisible by 3 because 18 ÷ 3 equals an integer. On the other hand, you would say that 12 is NOT divisible by 8, because 12 ÷ 8 is not an integer.