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GRE GRE Test Exam Questions
Graduate Record Examination Test: Verbal, Quantitative, Analytical Writing (Page 11 )

Updated On: 9-Mar-2026
Page 11 of 119
PDF with 793 Questions



If is an integer, which of the following numbers must also be an integer? (Choose all that apply.)

  1. x - 1
  2. x + 1



Answer(s): A,C,D

Explanation:

x - 1 is a multiple of 4.
x - 1 = 4k for some integer k.
Option x - 1
Since x - 1 = 4k, which is a multiple of 4, it must be an integer.
Option x + 1
Rewriting x + 1 in terms of k:

x + 1 = (x - 1) + 2 = 4k + 2
Since 4k is a multiple of 4 but adding 2 does not ensure divisibility by 4, this may not always be an integer.
Others options:



2k and 2k + 1 must be an integer.
The last option is not necessarily an integer.



If x is an integer and the sides of a triangle are x + 3, 2x, and x + 5, respectively, which of the following could NOT be the perimeter of the triangle?

  1. 16
  2. 20
  3. 28
  4. 30
  5. 32

Answer(s): D

Explanation:

The perimeter of the triangle is:
(x + 3) + (2x) + (x + 5) = 4x + 8.
Since the perimeter is 4x + 8, we check which given perimeter values are possible by solving for x.
For perimeter 16:
4x + 8 = 16
4x = 8
x = 2
For perimeter 20:
4x + 8 = 20
4x = 12
x = 3
For perimeter 28:
4x + 8 = 28
4x = 20
x = 5

For perimeter 30:
4x + 8 = 30
4x = 22
x = 5.5
For perimeter 32:
4x + 8 = 32
4x = 24
x = 6
The only perimeter value that is not possible because it results in a non-integer x is 30.





If P and Q are positive integers that have no common factors other than 1, what is the value of Q?

  1. 30
  2. 60
  3. 120
  4. 240
  5. 360

Answer(s): B

Explanation:

The denominators of the fractions are 2, 3, 4, 5, and 6. The least common denominator (LCD) of these numbers is the least common multiple (LCM) of 2, 3, 4, 5, and 6.
The prime factorizations are:
2 = 2

3 = 3

4 = 22

5 = 5

6 = 2 × 3

The LCM is the product of the highest powers of all prime factors:
LCM(2, 3, 4, 5, 6) = 22 × 3 × 5 = 60.



The result is already in its simplest form because 23 is a prime number, and it has no common factors with 60.





Let Q, R, and S be positive integers such that S is a factor of Q and R. Let



Which of the following is equal to ?

  1. |q - r|2
  2. |(q - r)(S)|2
  3. |q2 - r2|
  4. |(q2 - r2)(S)|
  5. |(q2 - r2)(S2)|

Answer(s): D

Explanation:

We are given that S is a factor of both Q and R, meaning that we can express Q and R in terms of S as:
Q = qS

R = rS

where q and r are integers.
We need to simplify the given expression:



Since Q = qS and R = rS,



S(q2 - r2), or |(q2 - r2)(S)|



The integer p is a prime number greater than 5, and 5 is a factor of p +p2.
What is the remainder when p is divided by 5?

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

Answer(s): E

Explanation:

We can express p + p2 as:
p + p2 = p(1 + p)
The problem states that 5 is a factor of p + p2, which means:
p(1 + p) 0 (mod 5)
This implies that p(1 + p) is divisible by 5. Since p is a prime number greater than 5, p is not divisible by 5.
Therefore, the factor of 5 must come from 1 + p.
We know that 1 + p must be divisible by 5. Thus, we have:
1 + p 0 (mod 5)
p -1 (mod 5)
Since -1 4 (mod 5), we conclude that:
p 4 (mod 5)



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