GRE Quantitative Basics: Number Properties

Published by Anirban Bose on

GRE Quantitative Basics: Number Properties

Roughly 25% of all questions on the GRE quantitative section directly or indirectly test your understanding of numbers and basic arithmetic.

Let’s start with numbers

INTEGERS

INTEGERS are the numbers 1, 2, 3, …, together with their negatives, −1, −2, −3, …, and 0. The set of integers 1, 2, 3, … are called positive integers and the set −1, −2, −3, … are called negative integers.

0 is neither positive nor negative.

HINT: on the GRE quantitative section, if a question mentions ‘all non-negative integers’ it means 0 and positive integers. Similarly, ‘all non-positive integers’ refers to 0 and negative integers.

Thus, the set of integers is {…, −3, −2, −1, 0, 1, 2, 3, …}.

When integers are added, subtracted, or multiplied, the result is always an integer. However, division of integers can yield integers or decimals. More on that later.

NATURAL NUMBERS are also known as COUNTING NUMBERS and basically represent numbers used for counting. Because we do not start count at zero and do not count in negatives, natural numbers start from 1. The set {1, 2, 3 ….} is the set of natural numbers.

WHOLE NUMBERS include 0 and natural numbers, the set {0, 1, 2, 3 ….} is the set of whole numbers.

Let’s consider it on the number line below

Numbers on the number line increase as you go from left to right, so 3 is greater than 1 but -1 is greater than -3.

INTEGER ARITHMETIC

If we consider the arithmetic operators of addition, subtraction, multiplication and division we observe the following

The product of two positive integers is a positive integer.   3 x 6 = 18

The product of two negative integers is a positive integer.  -3 x -6 = 18

The product of a positive integer and a negative integer is a negative integer.

-3 x 6 = -18 and 3 x -6 = -18

The summation of two negative integers is a smaller negative integer (-3) + (- 4) = -7

The subtraction of two negative integers could be positive or negative.

(-3) – (- 4) = -3+4 = 1

(-4) – (- 3) = -4 + 3 = – 1

Example Question 1:

If a is a positive integer and b is a negative integer, then which of the following must be positive?

Select all that apply.

  1. ab +10
  2. -20(ab)
  3. (-a+10)(-b+10)
  4. -(a+3)b
SOLUTION

Instead of trying to work the question, let us work the answer options

WARNING: Do NOT plug in values for a and b. ‘Plugging in’ is at best an inefficient strategy, it requires you to plug in multiple values at critical limits and cross check the answer. At worst, plugging in is totally erroneous. There are a few exceptions where plugging in works but they are few and far between.

ab is a negative number so ab+10 may or may not be positive depending on the value of ab. If ab = -8 then ab+10 >0 but if ab = -14 then ab+10 <0.

If in this example you had plugged in a = 2 and b= -3 then ab+10 would most definitely be positive but that’s not always the case. So you see why ‘plugging in’ is not a wise approach.

-20(ab) is the multiplication of 2 negative numbers, -20 and ab. It is definitely positive.

Consider separately the two multiplicands of (-a+10)(-b+10). (-b+10) is definitely positive but (-a+10) may or may not be positive so the product is not necessarily always positive.

Consider separately the three multiplicands of -(a+3)b. b is negative, (a+3) is positive so (a+3)b is negative. Multiply a negative number by -1 and you get a positive number.

So

  • -20(ab)
  • -(a+3)b

must be positive.

The GRE quantitative section often tests on on such simple concepts. Questions involving advanced concepts or complex calculation seldom make it to the GRE.

ODD & EVEN

Now let us look at another important property of numbers. If an integer is divisible by 2, it is called an EVEN integer; otherwise, it is an ODD integer. Note that when an odd integer is divided by 2, the remainder is always 1. The set of even integers is {…, −6, −4, −2, 0, 2, 4, 6, …}, and the set of odd integers is {…, −5, −3, −1, 1, 3, 5, …}.

NOTE: Negative integers are also odd or even. 0 is an even number because 0 divided by 2 leaves 0 as remainder.

The sum of two even integers is an even integer.

2+4 = 6, 10 + 18 = 28

The sum of two odd integers is an even integer.

3+5 = 8, 11 + 19 = 30

The sum of an even integer and an odd integer is an odd integer. 3 +8 = 11, 12 + 7 = 19

The product of two even integers is an even integer.

4 x 8 = 32, 12 x 16 = 192

The product of two odd integers is an odd integer. 

3 x 9 = 27, 15 x 17 = 255

 The product of an even integer and an odd integer is an even integer.

4 x 9 = 36, 17 x 6 = 102

ODD-EVEN RULES

These might appear to be a lot of rules but it is really very simple when you consider the following: Odd integers have a remainder of 1 and even integers have 2 as a factor.

EVEN +EVEN has a common factor of 2, so it is EVEN

ODD + EVEN does not have a factor of 2, so it is ODD

Now, ODD + ODD has two 1s as remainders which join to make a 2, so it is EVEN

Clearly, EVEN x EVEN has a common factor of 2, so it is EVEN

And, EVEN x ODD has a common factor of 2, so it is EVEN

ODD x ODD does not have any factor of 2, so it is ODD

This is one of those cases where ‘plugging in’ will help. Plug in any even and odd value and whatever the result of the computation, it must be true for all computations.

2+3=5 so all EVEN + ODD = ODD

2 x 3 = 6 so all EVEN x ODD = EVEN, and so on.

Example Question 2:

If a is an even integer and b is an odd integer such that both a and b are positive, then which of the following is even?

Select all that apply.

  1. ab+1
  2. ab+2
  3. (a+1)(b-1)
  4. a(b+3)
SOLUTION

ab+1 : ab is even, ab+1 is odd

ab+2 : ab is even, ab+2 is also even

(a+1)(b-1) : a+1 is odd and b – 1 is even, odd x even is even

a(b+3) : a is even and b + 3 is odd, even x odd is even

What you just worked on was an easy question, let’s look at another GRE Quantitative question on odd/even by complicating the previous question

Example Question 3:

If a is an even integer and b is an odd integer such that ab>0, then which of the following is an even positive integer.

Select all that apply.

  1. ab+1
  2. ab+2
  3. (a+3)(b-3)
  4. a(b+3)

What’s the complication: ab>0 means either a and b are both positive or a and b are both negative

SOLUTION

ab+1 : ab is even, ab+1 is odd

ab+2 : ab is even, ab+2 is also even and is also positive

(a+3)(b-3) : a + 3 is odd (positive or negative) and b – 3 is even (positive or negative), odd x even is even but we can’t be sure if it is positive.

If a = 4 and b = 5 then (a+3)(b-3) = 7(2)= 14 but if a = -2 and b = -5 then (a+3)(b-3) = 1(-8)=-8 .

a(b+3) : a is even and b + 3 is odd but can be positive or negative depending on the absolute value of b. So,  even x odd is even but could be positive or negative.

So, answer is: ONLY ab + 2

FACTORS

When integers are multiplied, each of the multiplied integers is called a factor or divisor of the resulting product. For example, (2)(3)(10) = 60, so 2, 3, and 10 are factors of 60.

Any number that divides 60 without remainder is a factor of 60, so the integers 4, 15, 5, and 12 are also factors of 60. Of course, 60 is also a factor of 60 and so is 1.

NOTE:  The positive factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The negatives of these integers are also factors of 60.

The positive factors of 10 are 1, 2, 5 and 10

The negative factors of 10 are -1, -2, -5, and -10.

Let’s use these concepts to work on another simple question. The following question is simple in the concepts it tests but is of medium level of difficulty in that it has a lot of trick options. You will encounter such simple but tricky questions on the GRE quantitative section.

Example Question 4:

The square of an integer x is less than 20. How many possible values are there for x?

A. Four

B. Five

C. Eight

D. Nine

E. Twenty

If the square of x is less than 20 and x is an integer then the max value of x is 4 because the square of 5 is 25 (greater than 20). So x can take the values of 1, 2, 3 and 4.

Does that make option A your answer? NO.

Don’t forget x could also be -1, -2, -3, or -4.

Does that make option C your answer? NO.

Don’t forget x could also be 0. So x can be {-4, -3, -2, -1, 0, 1, 2, 3, 4}.

Answer: Option D, Nine

PRIMES & COMPOSITES

Now that we understand factors, let’s look at primes and composites.

The GRE Quantitative Review defines primes and composites in the following manner.

 A prime number is an integer greater than 1 that has only two positive divisors: 1 and itself.

An integer greater than 1 that is not a prime number is called a composite number.

A better way to look at primes and composites would be

A PRIME number is any number that has exactly 2 distinct factors (or divisors)

A COMPOSITE number is any number that has at least 3 distinct factors (or divisors)

NOTE: The categorization of prime or composite only applies to positive integers. Zero and negative integers or non-integers are not classified as prime or composite.

USEFUL INFO

1 has only one distinct factor, the number 1. So 1 is neither prime nor composite. 1 is categorized as UNIQUE.

2 has exactly 2 divisors, 1 and 2 so it is prime. 2 is the first prime number. 2 is also the only prime number that is even, all other prime numbers are odd.

It is easy to understand intuitively that any even number greater than 2 will have at least 2 even factors 2 and the number itself, it will also have 1 as a factor. Therefore, it must have at least 3 factors and is hence composite.

3 has exactly 2 divisors, 1 and 3 so it is prime.

4 has 3 divisors, 1, 2 and 4, so it is composite. 4 is the first composite number.

NOTE:

The first ten prime numbers lie between 1 and 30 and they are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

There are 25 prime numbers in the first 100 natural numbers and they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

This information is quite useful in solving the next question

Example Question 5:

Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given. Select one of the following four answer choices.

A Quantity A is greater.

B Quantity B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

A symbol that appears more than once in a question has the same meaning throughout the question.

P is number of prime numbers less than 50.

Q is the number of prime numbers greater than 50 but less than 100.

Quantity A                                                                          Quantity B

P                                                                                                      Q

SOLUTION

There are 2 ways of looking at the question.

1: There are 25 prime numbers in the first 100 natural numbers and 10 are less than 30. So 31, 37, 41, 43, and 47 are the additional 5 primes, bringing the total number of primes less than 50 to 15. Obviously, the remaining 10 primes lie between 50 and 100.

P > Q

2: There are 4 prime numbers in the first 10 natural numbers and they are 2, 3, 5, and 7. Similarly, there are 4 prime numbers between 10 and 20 and they are 11, 13, 17, and 19. But only 2 primes between 20 and 30 (23 and 29) and 2 between 30 and 40 (31 and 37) so the number of primes keep decreasing. This is also intuitive, as the numbers increase in magnitude their chances of having 3 divisors also increases.

P > Q

  Answer: Option A, quantity A is greater than quantity B                      

QUICK SUMMARY

Remember, the GRE Quantitative section is not difficult. It tests you on math basics and uses conceptual weakness to lead you to the wrong answer. Get your basics right and you will ace the section.

A helpful way of remembering all this information is through this picture below.

Happy Prepping!


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