  # Integers

All GMAT exams include a number of questions that deal with integers. Which topics are covered?

• Properties of integers
• Divisibility rules
• Factors and prime factors
• Arithmetic sets
• Remainders
• Rules of odds and evens

On this page you can view examples of these types of questions, and also login to our advanced on-line practice software to practice free integer questions.
Here's an example of this type of question in the quantitative section of the GMAT:

Which of the following is an integer?
• √5728
• √5929
• √3543
• √2792
• √8367

Solution:
We are asked which of the numbers presented is an integer. Of course, we cannot use a calculator in the GMAT.  However, in this case, there is no need to calculate and the question can easily solved without determining the root. How? When we square an integer, the unit's digit cannot be 2, 3, 7 or 8. Therefore options A, C, D and E can be ruled out immediately.

Here's another integers question, this time in the DS format:

x, y, z, m and n are consecutives integers (not necessarily in that order). Is x=0?

(1) x+n=0
(2) y+z+m=0

(A)   Statement (1) ALONE is sufficient to answer the question, but statement (2) ALONE is not.
(B)    Statement (2) ALONE is sufficient to answer the question, but statement (1) ALONE is not.
(C)    Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement alone is sufficient.
(D)   Either statement by itself is sufficient to answer the question.
(E)    Statements (1) and (2) taken together are not sufficient to answer the question, and additional data are needed to answer the question.

Solution:

We are told that the integers are consecutive. We are asked if x is equal to 0. As we know, since this is a data sufficiency question, ruling out the possibility that x is equal to 0 would be sufficient to answer the question.

The first statement tells us that 0=n+x. Two integers added together can only be equal to 0 if both numbers are 0 or if they are opposite numbers (for example 1 and -1). In this case, both numbers cannot be equal to 0 since we know that they are consecutive, therefore x cannot equal 0. This means that the statement is sufficient to answer the question.

The second statement tells us that 0=y+z+m. This can only be the case in one of the following scenarios:
• If all three equal 0. This cannot be so here since we know they are consecutive numbers, i.e; they are definitely different numbers.
• If the total of two of the numbers is equal to the opposite of the third number, for example, 1, 2 and -3. This cannot be the case because we only have 5 numbers.
• If two of them are opposite numbers and the third is equal to 0. This must be the case in this instance.
If so, we know for certain that one of the three numbers y, z or m is equal to 0. Since they are consecutive numbers, we know for certain that x is not equal to 0. This means that the second statement is also sufficient to answer the question.