Integers

Concepts

Integers Extended Concept

1. Definition:

  • Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. The set of integers is denoted by ( \mathbb{Z} ) and includes: (, -3, -2, -1, 0, 1, 2, 3, ).

2. Properties of Integers:

  • Addition and Subtraction:
  • The sum or difference of two integers is always an integer.
  • Example: ( 5 + (-3) = 2 ), ( -4 – 6 = -10 ).
  • Multiplication:
  • The product of two integers is always an integer.
  • Example: (   ( = -12 ), ( -2  (-5) = 10 ).
  • Division:
  • The quotient of two integers is not always an integer.
  • Example: ( = 2 ) (integer), but ( = 3.5 ) (not an integer).

3. Even and Odd Integers:

  • Even Integers: Divisible by 2. Examples: ( -4, -2, 0, 2, 4,  ).
  • Odd Integers: Not divisible by 2. Examples: (-3, -1, 1, 3, 5,).

4. Positive and Negative Integers:

  • Positive Integers: Greater than zero. Examples: ( 1, 2, 3, \ldots ).
  • Negative Integers: Less than zero. Examples: ( -1, -2, -3, \ldots ).

5. Zero (0):

  • Zero is an integer that is neither positive nor negative.
  • It is an even number.

6. Absolute Value:

  • The absolute value of an integer is its distance from zero on the number line, regardless of direction.
  • Denoted by ( |a| ).
  • Example: ( |3| = 3 ), ( |-3| = 3 ).

7. Number Line:

  • Integers are represented on a number line with zero at the center, positive integers to the right, and negative integers to the left.

8. Properties of Operations:

  • Commutative Property:
  • Addition: ( a + b = b + a )
  • Multiplication: ( a \times b = b \times a )
  • Associative Property:
  • Addition: ( (a + b) + c = a + (b + c) )
  • Multiplication: ( (a \times b) \times c = a \times (b \times c) )
  • Distributive Property:
  • ( a \times (b + c) = (a \times b) + (a \times c) )

9. Practical Applications:

  • GRE Problems: Integers are used in various types of GRE questions, including algebra, number properties, and word problems. Understanding their properties helps in solving these problems efficiently.

Absolutely! Here are some GRE-style practice questions focused on integers:

Practice Questions

1. If ( x ) and ( y ) are integers such that ( x ) is positive and ( y ) is negative, which of the following must be true?

  • A) ( x + y ) is positive
  • B) ( x \times y ) is positive
  • C) ( x – y ) is positive
  • D) ( x \div y ) is an integer

2. Which of the following expressions always results in an even number?

  • A) ( 3a + 4b ) where ( a ) and ( b ) are integers
  • B) ( a2 + b2 ) where ( a ) and ( b ) are even integers
  • C) ( 2a + 3b ) where ( a ) and ( b ) are integers
  • D) ( a^2 – b^2 ) where ( a ) and ( b ) are odd integers

3. If ( n ) is an integer, which of the following is true about ( n^2 )?

  • A) ( n^2 ) is always positive
  • B) ( n^2 ) is always even
  • C) ( n^2 ) is always odd
  • D) ( n^2 ) is always non-negative

4. Given that ( a ) and ( b ) are integers, which of the following statements is true?

  • A) ( a + b ) is always even
  • B) ( a \times b ) is always even
  • C) ( a – b ) is always odd
  • D) ( a \div b ) is not always an integer

5. If ( x ) is an even integer and ( y ) is an odd integer, which of the following is always true?

  • A) ( x + y ) is even
  • B) ( x – y ) is odd
  • C) ( x \times y ) is odd
  • D) ( x \div y ) is an integer

Answers:

  1. C) ( x – y ) is positive
  2. B) ( a^2 + b^2 ) where ( a ) and ( b ) are even integers
  3. D) ( n^2 ) is always non-negative
  4. D) ( a \div b ) is not always an integer
  5. B) ( x – y ) is odd

Class Materials

Topics 3

Positive, Negative & Consecutive Integer

A positive number is a real number that is greater than zero. A negative number is a real number that is smaller than zero.

Zero is not positive, nor negative.
product of the same sign is + ve and the product of the different sign is – ve

Consecutive Integers

Consecutive integers are integers that follow one another, without skipping any integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers.

[ii] If n is odd, the sum of consecutive integers is always divisible by n. Given, we have n = 4 consecutive integers.

The sum of 9+10+11=30, therefore, is divisible by 3.

[iii] If n is even, the sum of consecutive integers is never divisible by n. Given {9, 10, 11}, we have n= 4 consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

[iv] The product of n consecutive integers is always divisible

by n! or n.

Given n = 4 consecutive integers: {3, 4, 5, 6}. The product of 3×4×5×6 is 360, which is divisible by 4! =24.

PS Questions

1. List K consists of 12 consecutive integers, if -4 is the least integer in list K, what is the range of the positive integers in list K?

A.6

B. 11

C.7

D.12

E.5

2. If x is the sum of six consecutive integers, then x is divisible by which of the following:

I.3

II. 4.

III. 6

A. I only

B. II only

C.II only

D.I and III

E.I, II, and III

3. a, b, c, and d are consecutive integers, and a < b < c < d. If the product of b, c, and d is twice that of a, b, and c, then bc =

A.2

B. 6

C.12

D.20

E.30

4. How many integers are there between 51 and 107, inclusive?

A.51

B.55     C.56     D.57      E.58.

5. If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?

A. 9

B. 10

C. 11

D.12

E.13

6. If xyz < 0 and yz > 0, which of the following must be positive?

(A) xy

(B) xz

(C) (x2)yz

(D) x(y2)z

(E) xy(z2)

7. If 5,400n is the square of an integer, what is the smallest possible integer value of n?

(A) 2

(B) 3

(C) 5

(D) 6

(E) 15

8. If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

A. 2

B. 5

C. 6

D. 7

E. 14

9. If n and y are positive integers and 450y=n3, which of the following must be an integer?

I.y/(3×22×5) II.y/(32×2×5) III. y/(3×2×52 )
A None B I only C II only D III only E I, II, and III

1o. If a and b are negative odd numbers, and c and d are positive even numbers, which of the following is a possible value for?

A.-12 B.-1/2 C.3/4 D.13 E.34

Quantity Comparison Questions

11. a, b, c, and d are consecutive integers such that a < b < c < d

The average of a, b, c, and dThe average of b and c
  1. ab > 0,bc < 0
ac0
  1. mn < 0 ,mp > 0
np0
  1. abc < 0 ,b2c > 0
ab0
  1. a, b, and c are integers such that a < b < c
[a +b+c]/3b
  1. x, y, and z are integers ,xyz < 0 , yz < 0 , y < 0
xz
  1. The positive integer a is divisible by 2, and 0 < ab < 1
b1/2
  1. The sum of four consecutive integers is -2
The smallest of the four integers-2
  1. w, x, y, and z are consecutive positive integers and w<x<y<z.
The remainder when  (w +x)(x + y)(y + z) is divided by 21

20.m and n are positive integers.

m + nmn

Class Slide

Time-bound Tests

  • NP Integers Test – 1
  • NP Integers Test – 2
  • NP Integers Test – 3
  • NP Integers Test – 4
[iframe height=’600′ ]https://gre.mksprep.com/qnp/GRE_Integer_(M1)_Test_-_1/res/index.html[/iframe]
[iframe height=’600′ ]https://gre.mksprep.com/qnp/GRE_Integer_(M1)_Test_-_2/res/index.html[/iframe]
[iframe height=’600′ ]https://gre.mksprep.com/qnp/GRE_Integer_(M1)_Test_-_3/res/index.html[/iframe]
[iframe height=’600′ ]https://gre.mksprep.com/qnp/GRE_Integer_(M2)_Test_-_4/res/index.html[/iframe]