1. |21 2 2| 2 |5 2 6| 2 |23 1 4| 5
(A) 25
(B) 23
(C) 1
(D) 3
(E) 5
The correct answer is (C). First, determine each of the three absolute values:
|21 2 2| 5 |23| 5 3
|5 2 6| 5 |21| 5 1
|23 1 4| 5 |1| 5 1
Then combine the three results: 3 2 1 2 1 5 1.
Because multiplication (or division) involving two negative terms always results in a
positive number:
• Multiplication or division involving any even number of negative terms gives you
a positive number.
• Multiplication or division involving any odd number of negative terms gives you
a negative number.
2. A number M is the product of seven negative numbers, and the number N
is the product of six negative numbers and one positive number. Which of
the following holds true for all possible values of M and N ?
I. M 3 N , 0
II. M 2 N , 0
III. N 1 M , 0
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III
The correct answer is (C). The product of seven negative numbers is always a
negative number. (M is a negative number.) The product of six negative numbers
is always a positive number, and the product of two positive numbers is always a
positive number. (N is a positive number.) Thus, the product of M and N must be
a negative number; I is always true. Subtracting a positive number N from a
negative number M always results in a negative number less than M; II is always
true. However, whether III is true depends on the values of M and N.If|N| . |M|,
then N 1 M . 0, but if |N| , |M|, then N 1 M , 0.
Integers and the Four Basic Operations
When you combine integers using a basic operation, whether the result is an odd
integer, an even integer, or a non-integer depends on the numbers you combined. Here
are all the possibilities:
Chapter 10: Math Review: Number Theory and Algebra 223
www.petersons.com
ADDITION AND SUBTRACTION
• Integer ± integer 5 integer
• Even integer ± even integer 5 even integer
• Even integer ± odd integer 5 odd integer
• Odd integer ± odd integer 5 even integer
MULTIPLICATION AND DIVISION
• Integer 3 integer 5 integer
• Integer 4 non-zero integer 5 integer, but only if the numerator is divisible by the
denominator (if the result is a quotient with no remainder)
• Odd integer 3 odd integer 5 odd integer
• Even integer 3 non-zero integer 5 even integer
• Even integer 4 2 5 integer
• Odd integer 4 2 5 non-integer
GRE questions that test you on these rules sometimes resemble algebra problems, but
they’re really not. Just apply the appropriate rule or, if you’re not sure of the rule,
plug in simple numbers to zero in on the correct answer.
3. If P is an odd integer and if Q is an even integer, which of the following
expressions CANNOT represent an even integer?
(A) 3P 2 Q
(B) 3P 3 Q
(C) 2Q 3 P
(D) 3Q 2 2P
(E) 3P 2 2Q
The correct answer is (A). Since 3 and P are both odd integers, their product
(3P) must also be an odd integer. Subtracting an even integer (Q) from an odd
integer results in an odd integer in all cases.
FACTORS, MULTIPLES, AND DIVISIBILITY
Figuring out whether one number (f) is a factor of another (n) is easy: Just divide n by
f. If the quotient is an integer, then f is a factor of n (and n is divisible by f). If the
quotient is not an integer, then f is not a factor of n, and you’ll end up with a
remainder after dividing. For example, 2 is a factor of 8 because 8 4 2 5 4, which is an
integer. On the other hand, 3 is not a factor of 8 because 8 4 3 5
8
3
,or2
2
3
, which is a
non-integer. (The remainder is 2.)
Remember these four basic rules about factors, which are based on the definition of
the term “factor”:
PART IV: Quantitative Reasoning224
www.petersons.com
Any integer is a factor of itself.
1 and 21 are factors of all integers.
The integer zero has an infinite number of factors but is not a factor of any
integer.
A positive integer’s greatest factor (other than itself) will never be greater than
one half the value of the integer.
On the flip side of factors are multiples. If f is a factor of n, then n is a multiple of f.
For example, 8 is a multiple of 2 for the same reason that 2 is a factor of 8—because
8 4 2 5 4, which is an integer.
As you can see, factors, multiples, and divisibility are simply different aspects of the
same concept. So a GRE question about factoring is also about multiples and divis-
ibility.
4. If n . 6, and if n is a multiple of 6, which of the following is always a
factor of n?
(A) n 2 6 (B) n 1 6
(C)
n
3
(D)
n
2
1 3 (E)
n
2
1 6
The correct answer is (C). Since 3 is a factor of 6, 3 is also a factor of
any positive-number multiple of 6. Thus, if you divide any multiple of 6 by
3, the quotient will be an integer. In other words, 3 will be a factor of that
number (n). As for the incorrect choices, n 2 6, choice (A), is a factor of n
only if n 5 12. n 1 6, choice (B), can never be a factor of n because n 1 6
is greater than n. You can eliminate choices (D) and (E) because the
greatest factor of any positive number (other than the number itself) is
half the number, which in this case is
n
2
.
PRIME NUMBERS AND PRIME FACTORIZATION
A prime number is a positive integer greater than one that is divisible by only two
positive integers: itself and 1. Just for the record, here are all the prime numbers less
than 50:
2, 3, 5, 7
11, 13, 17, 19
23, 29
31, 37
41, 43, 47
The GRE might test you directly on prime numbers by asking you to identify all prime
factors of a number. These questions tend to be pretty easy.
Chapter 10: Math Review: Number Theory and Algebra 225
www.petersons.com
5. Column A Column B
The product of all different
prime-number factors of 42
42
(A) The quantity in Column A is greater.
(B) The quantity in Column B is greater.
(C) The quantities are equal.
(D) The relationship cannot be determined from the information given.
The correct answer is (C). The prime-number factors of 42 include 2, 3, and 7.
Their product is 42.
To find what’s called the prime factorization of a non-prime integer, divide the number
by the primes in order and use each repeatedly until it is no longer a factor. For
example:
110 5 2 3 55
5 2 3 5 3 11. This is the prime factorization of 110.
Stop when all factors are prime and then if a factor occurs more than once, use an
exponent to indicate this (i.e., write it in exponential form.)
6. Which of the following is a prime factorization of 144?
(A) 2
4
3 3
2
(B) 4 3 3
3
(C) 2
3
3 2 3 9
(D) 2
2
3 3 3 5
(E) 2 3 3
2
3 4
The correct answer is (A). Divide 144 by the smallest prime, which is 2.
Continue to divide the result by 2, then 3, and you ultimately obtain a prime-
number quotient:
144 5 2 3 72
5 2 3 2 3 36
5 2 3 2 3 2 3 18
5 2 3 2 3 2 3 2 3 9
5 2 3 2 3 2 3 2 3 3 3 3
5 2
4
3 3
2
EXPONENTS (POWERS)
An exponent,orpower, refers to the number of times a number (referred to as the
base) is used as a factor. In the number 2
3
, the base is 2 and the exponent is 3. To
calculate the value of 2
3
, you use 2 as a factor three times: 2
3
5 2 3 2 3 2 5 8.
PART IV: Quantitative Reasoning226
TIP
For the GRE, memorize a ll the
prime numbers less than 50, so
you don’t have to take time
thinking about whether you
can factor them.
www.petersons.com
On the GRE, questions involving exponents usually require you to combine two or
more terms that contain exponents. To do so, you need to know some basic rules. Can
you combine base numbers—using addition, subtraction, multiplication, or division—
before applying exponents to the numbers? The answer depends on which operation
you’re performing.
Combining Exponents by Addition or Subtraction
When you add or subtract terms, you cannot combine bases or exponents. It’s as simple
as that.
a
x
1 b
x
Þ (a 1 b)
x
a
x
2 b
x
Þ (a 2 b)
x
If you don’t believe it, try plugging in a few easy numbers. Notice that you get a
different result depending on which you do first: combine bases or apply each
exponent to its base.
7. If x 522, then x
5
2 x
2
2 x 5
Enter a number in the box.
The correct answer is (34). You cannot combine exponents here, even though
the base is the same in all three terms. Instead, you need to apply each exponent,
in turn, to the base, then subtract:
x
5
2 x
2
2 x 5 (22)
5
2 (22)
2
2 (22) 5232 2 4 1 2 5234
Combining Exponents by Multiplication or Division
It’s a whole different story for multiplication and division. First, remember these two
simple rules:
You can combine bases first, but only if the exponents are the same:
a
x
3 b
x
5 (ab)
x
You can combine exponents first, but only if the bases are the same. When
multiplying these terms, add the exponents. When dividing them, subtract the
denominator exponent from the numerator exponent:
a
x
3 a
y
5 a
~x 1 y!
a
x
a
y
5 a
~x 2 y!
When the same base appears in both the numerator and denominator of a fraction,
you can cancel the number of powers common to both.
Chapter 10: Math Review: Number Theory and Algebra 227
TIP
For numeric-entry questions
(which might appear only o n
the computer-based version of
the GRE), you enter a
negative sign by typing the
hyphen key. T o erase the
negative sign, type the
hyphen key again.
www.petersons.com
8. Which of the following is a simplified version of
x
2
y
3
x
3
y
2
if x,y Þ 0?
(A)
y
x
(B)
x
y
(C)
1
xy
(D) 1 (E) x
5
y
5
The correct answer is (A). The simplest approach to this problem is to
cancel x
2
and y
2
from numerator and denominator. This leaves you with x
1
in the denominator and y
1
in the numerator.
“Canceling” a base’s powers in a fraction’s numerator and denominator is actually a
shortcut to applying the rule
a
x
a
y
5 a
(x 2 y)
along with another rule, a
2x
5
1
a
x
, that you’ll
review immediately ahead.
Additional Rules for Exponents
To cover all your bases, also keep in mind these three additional rules for exponents:
When raising an exponential number to a power, multiply exponents:
(a
x
)
y
5 a
xy
Any number other than zero (0) raised to the power of zero (0) equals 1:
a
0
5 1[a Þ 0]
Raising a base other than zero to a negative exponent is equivalent to 1 divided by
the base raised to the exponent’s absolute value:
a
2x
5
1
a
x
The preceding three rules are all fair game for the GRE. In fact, a GRE question
might require you to apply more than one of these rules.
9. (2
3
)
2
3 4
23
5
(A)
1
8
(B)
1
2
(C)
2
3
(D) 1 (E) 16
The correct answer is (D). ~2
3
!
2
3 4
23
5 2
~2!~3!
3
1
4
3
5
2
6
4
3
5
2
6
2
6
5 1.
PART IV: Quantitative Reasoning228
www.petersons.com
Exponents You Should Know
For the GRE, memorize the exponential values in the following table. You’ll be glad
you did, since these are the ones you’re most likely to see on the exam.
Power and Corresponding Value
Base 2345678
2 4 8 16 32 64 128 256
3 9 27 81 243
4 16 64 256
5 25 125 625
6 36 216
Exponents and the Real Number Line
Raising bases to powers can have surprising effects on the magnitude and/or sign—
negative vs. positive—of the base. You need to consider four separate regions of the
real number line:
Values greater than 1 (to the right of 1 on the number line)
Values less than 21 (to the left of 21 on the number line)
Fractional values between 0 and 1
Fractional values between 21 and 0
The next table indicates the impact of positive-integer exponent (x) on base (n)for
each region.
n . 1 n raised to any power: n
x
. 1 (the greater the exponent, the
greater the value of n
x
)
n ,21 n raised to even power: n
x
. 1 (the greater the exponent, the
greater the value of n
x
)
n raised to odd power: n
x
, 1 (the greater the exponent, the lesser
the value of n
x
)
0 , n , 1 n raised to any power: 0 , n
x
, 1 (the greater the exponent, the
lesser the value of n
x
)
21 , n , 0 n raised to even power: 0 , n
x
, 1 (the greater the exponent, the
lesser the value of n
x
, approaching 0 on the number line)
n raised to odd power: 21 , n
x
, 0 (the greater the exponent, the
greater the value of n
x
, approaching 0 on the number line)
When you apply the preceding set of rules to a GRE question, it can be surprisingly
easy to confuse yourself, especially if the question is designed to create confusion.
Here are two challenging examples.
Chapter 10: Math Review: Number Theory and Algebra 229
www.petersons.com
10. If 21 , x , 0, which of the following must be true?
I. x , x
2
II. x
2
, x
3
III. x , x
3
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
The correct answer is (D). The key to analyzing each equation is that raising
x to successively greater odd powers moves the value of x closer to zero (0) on the
number line, while raising x to an even power yields a positive value.
I must be true. Since x is given as a negative number, x
2
must be positive and
thus greater than x.
II cannot be true. Since x is given as a negative number, x
2
must be positive,
while x
3
must be negative. Thus, x
2
is greater than x
3
.
III must be true. Both x
3
and x are negative fractions between 0 and 21, but x
3
is
closer to zero (0) on the number line—that is, greater than x.
11. 0 , x , y
Column A
Column B
xy
xy
55
55
+
xy
xy
+
(
)
(
)
5
5
(A) The quantity in Column A is greater.
(B) The quantity in Column B is greater.
(C) The quantities are equal.
(D) The relationship cannot be determined from the information given.
The correct answer is (B). In Column B, the denominator can be expressed as
xy
55
. Since the denominators in the two quantities are the same, and since both
are positive in value, you can cancel them. You need only compare the two
numerators. If you mistakenly assume that (x + y)
5
=
xy
55
+
, you might select
answer choice (C). In fact, with x and y both greater than 1, you can easily see
that (x + y)
5
will always be greater than x
5
+ y
5
by considering the result of
combining five binomials—(x + y)(x + y)(x + y)(x + y)(x + y)—by multiplication.
ROOTS AND RADICALS
On the flip side of exponents and powers are roots and radicals. The square root of a
number n is a number that you “square” (multiply by itself, or raise to the power of 2)
to obtain n.
2 5
=
4 (the square root of 4) because 2 3 2(or2
2
) 5 4
PART IV: Quantitative Reasoning230
ALERT!
You can cancel denominators
across columns only if you are
certain that Quantity A and
Quantity B are either both
positive or both negative.
www.petersons.com
The cube root of a number n is a number that you raise to the power of 3 (multiply by
itself twice) to obtain n. You determine greater roots (for example, the “fourth root”) in
the same way. Except for square roots, the radical sign will indicate the root to be
taken.
2 5
=
3
8 (the cube root of 8) because 2 3 2 3 2(or2
3
) 5 8
2 5
=
4
16 (the fourth root of 16) because 2 3 2 3 2 3 2(or2
4
) 5 16
For the GRE, you should know the rules for simplifying and combining radical
expressions.
Simplifying Radicals
On the GRE, always look for the possibility of simplifying radicals by moving what’s
under the radical sign to the outside of the sign. Check inside your square-root
radicals for perfect squares: factors that are squares of nice tidy numbers or other
terms. The same advice applies to perfect cubes, and so on.
=
4a
2
5 2|a|
4 and a
2
are both perfect squares; remove them from under the radical sign,
and find each one’s square root.
=
8a
3
5
=
~4!~2!a
3
5 2a
=
2a
8 and a
3
are both perfect cubes, which contain perfect-square factors;
remove the perfect squares from under the radical sign, and find each one’s
square root.
You can simplify radical expressions containing fractions in the same way. Just be
sure that what’s in the denominator under the radical sign stays in the denominator
when you remove it from under the radical sign. For example:
For all non-negative values of x,
Î
20x
x
2
5
Î
~4!~5!
x
2
5
2
=
5
x
2
Here’s another example:
Î
3
3
8
5
Î
3
3
2
3
5
1
2
=
3
3
12. For all non-negative values of a and b,
Î
28a
6
b
4
36a
4
b
6
5
(A)
a
b
Î
a
2b
(B)
a
2b
Î
a
b
(C)
a
3b
=
7
(D)
a
2
3b
2
=
2 (E)
2a
3b
The correct answer is (C). Divide a
4
and b
4
from the numerator and
denominator of the fraction. (In other words, factor them out.) Also, factor
out 4 from 28 and 36. Then, remove perfect squares from under the radical
sign:
Chapter 10: Math Review: Number Theory and Algebra 231
TIP
Whenever you see a non-
prime number under a square-
root radical sign, factor it to
see whether it contains
perfect-square factors you
can move outside the radical
sign. More than likely, you
need to do so to solve the
problem at hand.
www.petersons.com
Î
28a
6
b
4
36a
4
b
6
5
Î
7a
2
9b
2
5
a
=
7
3b
or
a
3b
=
7
In GRE questions involving radical terms, you may want to remove a radical term
from a fraction’s denominator to match the correct answer. To accomplish this, mul-
tiply both numerator and denominator by the radical value. This process is called
“rationalizing the denominator.” Here’s an example of how to do it:
3
=
15
5
3
=
15
=
15
=
15
5
3
=
15
15
or
1
5
=
15
Combining Radical Terms
The rules for combining terms that include radicals are similar to those for exponents.
Keep the following two rules in mind; one applies to addition and subtraction, while
the other applies to multiplication and division.
ADDITION AND SUBTRACTION
If a term under a radical is being added to or subtracted from a term under a different
radical, you cannot combine the two terms under the same radical.
=
x 1
=
y Þ
=
x 1 y
=
x 2
=
y Þ
=
x 2 y
=
x 1
=
x 5 2
=
x not
=
2x
On the GRE, if you’re asked to combine radical terms by adding or subtracting,
chances are you’ll also need to simplify radical expressions along the way.
13.
=
24 2
=
16 2
=
6 5
(A)
=
6 2 4
(B)
4 2 2
=
2
(C) 2
(D)
=
6
E.
2
=
2
The correct answer is (A). Although the numbers under the three radicals
combine to equal 2, you cannot combine terms this way. Instead, simplify the first
two terms, then combine the first and third terms:
=
24 2
=
16 2
=
6 5 2
=
6 2 4 2
=
6 5
=
6 2 4
PART IV: Quantitative Reasoning232
www.petersons.com
. power: n
x
. 1 (the greater the exponent, the
greater the value of n
x
)
n raised to odd power: n
x
, 1 (the greater the exponent, the lesser
the value of. n
x
, 1 (the greater the exponent, the
lesser the value of n
x
)
21 , n , 0 n raised to even power: 0 , n
x
, 1 (the greater the exponent, the
lesser the value