Tài liệu MATH REVIEW for Practicing to Take the GRE General Test pdf

ARITHMETIC

1.1 Integers

The set of integers, I, is composed of all the counting numbers (i.e., 1, 2,
3, . . .), zero, and the negative of each counting number; that is,
I = , , , , , , , , .3 2 10123
:?

Therefore, some integers are positive, some are negative, and the integer 0 is
neither positive nor negative. Integers that are multiples of 2 are called even
integers, namely
, , , , , , , , 6420246
:?
All other integers are called
odd integers; therefore
, , , , , , , 531135
:?
represents the set of all
odd integers. Integers in a sequence such as 57, 58, 59, 60, or
−
14,
−
13,
−
12,
−
11
are called consecutive integers.
The rules for performing basic arithmetic operations with integers should be
familiar to you. Some rules that are occasionally forgotten include:
(i) Multiplication by 0 always results in 0; e.g., (0)(15) = 0.
(ii) Division by 0 is not defined; e.g., 5 ÷ 0 has no meaning.
(iii) Multiplication (or division) of two integers with different signs yields
a negative result; e.g.,
((8)-=-7) 56 and ()()  =-12 4 3
(iv) Multiplication (or division) of two negative integers yields a positive
result; e.g.,
()( ) =512 60 and ()() - =24 38
The division of one integer by another yields either a zero remainder, some-
times called “dividing evenly,” or a positive-integer remainder. For example,
215 divided by 5 yields a zero remainder, but 153 divided by 7 yields a remain-
der of 6.

5 215
20
43
15
15

7 153
14
21
13
7

0= Remainder 6 =
Remainder
When we say that an integer N is divisible by an integer x, we mean that N
divided by x yields a zero remainder.
The multiplication of two integers yields a third integer. The first two integers
are called factors, and the third integer is called the product. The product is said
to be a multiple of both factors, and it is also divisible by both factors (providing
the factors are nonzero). Therefore,
since
()() ,27 14= we can say that
2 and 7 are factors and 14 is the product,
14 is a multiple of both 2 and 7,
and 14 is divisible by both 2 and 7.
Whenever an integer N is divisible by an integer x, we say that x is a divisor
of N. For the set of positive integers, any integer N that has exactly two distinct
positive divisors, 1 and N, is said to be a prime number. The first ten prime
numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
The integer 14 is not a prime number because it has four divisors: 1, 2, 7, and 14.
The integer 1 is not a prime number because it has only one positive divisor.
6


1.2 Fractions

A fraction is a number of the form
a
b
,
where a and b are integers and b

0.
The a is called the numerator of the fraction, and b is called the denominator.
For example,
-7
5
is a fraction that has
-
7 as its numerator and 5 as its denomi-
nator. Since the fraction
a
b
means a

b, b cannot be zero. If the numerator
and denominator of the fraction
a
b
are both multiplied by the same integer,
the resulting fraction will be equivalent to
a
b
.
For example,
-
=
-
=
-7
5
7)
4
54
28
20
(()
()()
.

This technique comes in handy when you wish to add or subtract fractions.
To add two fractions with the same denominator, you simply add the
numerators and keep the denominator the same.
-
+=
-+
=
-8
11
5
11
8
5
11
3
11

If the denominators are not the same, you may apply the technique mentioned
above to make them the same before doing the addition.
5
12
2
3
5
12
24
34
5
12
8
12
58
12
13
12
+= + = + =
+
=
()()
()()

The same method applies for subtraction.
To multiply two fractions, multiply the two numerators and multiply the two
denominators (the denominators need not be the same).
10
7
1
3
10 1
7) 3
10
21




-




=
-
=
-
()()
(()

To divide one fraction by another, first invert the fraction you are dividing by,
and then proceed as in multiplication.
17
8
3
5
17
8
5
3
17) 5
3
85
24
=








==
(()
(8)( )

An expression such as
4
3
8
is called a mixed fraction; it means
4
3
8
+ .

Therefore,
4
3
8
4
3
8
32
8
3
8
35
8
=+ = + = .


7

then


1.3 Decimals

In our number system, all numbers can be expressed in decimal form using
base 10. A decimal point is used, and the place value for each digit corresponds
to a power of 10, depending on its position relative to the decimal point. For
example, the number 82.537 has 5 digits, where
“8” is the “tens” digit; the place value for “8” is 10.
“2” is the “units” digit; the place value for “2” is 1.
“5” is the “tenths” digit; the place value for “5” is
1
10
.

“3” is the “hundredths” digit; the place value for “3” is
1
100
.

“7” is the “thousandths” digit; the place value for “7” is
1
1000
.

Therefore, 82.537 is a short way of writing
(8)( ) ( )( ) ( ) ( ) ( ,10 2 1 5
1
10
3
1
100
7)
1
1000
++




+




+




or
80 + 2 + 0.5 + 0.03 + 0.007.
This numeration system has implications for the basic operations. For addi-
tion and subtraction, you must always remember to line up the decimal points:

126 5
68 231
194 731
.
.
.
+

126
5
68 231
58 269
.
.
.
-

To multiply decimals, it is not necessary to align the decimal points. To deter-
mine the correct position for the decimal point in the product, you simply add
the number of digits to the right of the decimal points in the decimals being mul-
tiplied. This sum is the number of decimal places required in the product.

15 381 3
14 2
61524
15381
2 15334
5
.( )
.( )
.
(
decimal places
decimal places
decimal places)


To divide a decimal by another, such as 62.744 ÷ 1.24, or
124 62744
,

first move the decimal point in the divisor to the right until the divisor becomes
an integer, then move the decimal point in the dividend the same number of
places;
124 6274.4
.

This procedure determines the correct position of the decimal point in the quo-
tient (as shown). The division can then proceed as follows:
8
0


124 6274
50
6
620
744
744
0
.4
.

Conversion from a given decimal to an equivalent fraction is straightforward.
Since each place value is a power of ten, every decimal can be converted easily
to an integer divided by a power of ten. For example,
84 1
841
10
917
917
100
0 612
612
1000
.
.
.
=
=
=

The last example can be reduced to lowest terms by dividing the numerator
and denominator by 4, which is their greatest common factor. Thus,
0612
612
1000
612 4
1000 4
153
250
.(==


= in lowest terms).
Any fraction can be converted to an equivalent decimal. Since the fraction
a
b

means
ab ,
we can divide the numerator of a fraction by its denominator to
convert the fraction to a decimal. For example, to convert
3
8
to a decimal, divide
3 by 8 as follows.
83000
0375
24
60
56
40
40
0
.
.


9


1.4 Exponents and Square Roots

Exponents provide a shortcut notation for repeated multiplication of a number
by itself. For example, “3
4
” means (3)(3)(3)(3), which equals 81. So, we say that
3
4
= 81; the “4” is called an exponent (or power). The exponent tells you how
many factors are in the product. For example,
2 22222 32
10 10 10 10 10 10 10 1 000 000
4 444 64
1
2
1
2
1
2
1
2
1
2
1
16
5
6
3
4
==
==
- = =-




=
















=
()()()()()
()()()()()() , ,
() ()()()

When the exponent is 2, we call the process squaring. Therefore, “5
2
” can be
read “5 squared.”
Exponents can be negative or zero, with the following rules for any nonzero
number m.
m
m
m
m
m
m
m
m
m
n
n
n
0
1
2
2
3
3
1
1
1
1
1
=
=
=
=
=
for all integers .

If m = 0, then these expressions are not defined.
A square root of a positive number N is a real number which, when squared,
equals N. For example, a square root of 16 is 4 because 4
2
= 16. Another square
root of 16 is –4 because (–4)
2
= 16. In fact, all positive numbers have two
square roots that differ only in sign. The square root of 0 is 0 because 0
2
= 0.
Negative numbers do not have square roots because the square of a real number
cannot be negative. If N > 0, then the positive square root of N is represented by
N
,
read “radical N.” The negative square root of N, therefore, is represented
by
-
N
.
Two important rules regarding operations with radicals are:
If a > 0 and b > 0, then
(i)
ab ab1616
= ;
e.g.,
5161 620 100 10
==

(ii)
a
b
a
b
= ;
e.g.,
192
4
== = =
48 16 3 16 3
4
3()()
1616

10


1.5 Ordering and the Real Number Line

The set of all real numbers, which includes all integers and all numbers with
values between them, such as 1.25,
2
3
2,,
etc., has a natural ordering, which
can be represented by the real number line:

Every real number corresponds to a point on the real number line (see examples
shown above). The real number line is infinitely long in both directions.
For any two numbers on the real number line, the number to the left is less
than the number to the right. For example,
-<-
-
<
<
5
3
2
175 2
5
2
71
.
.

Since 2 < 5, it is also true that 5 is greater than 2, which is written “5 > 2.”
If a number N is between 1.5 and 2 on the real number line, you can express
that fact as 1.5 < N < 2.
11


1.6 Percent

The term percent means per hundred or divided by one hundred. Therefore,
43%
43
100
0
300%
300
100
3
05%
05
100
0005
==
==
===
.43
.
.
.

To find out what 30% of 350 is, you multiply 350 by either 0.30 or
30
100
,

30% of 350 = (350) (0.30) = 105
or
30% of
350 = (350)
30
100




=




==()
,
.350
3
10
1050
10
105

To find out what percent of 80 is 5, you set up the following equation and
solve for x:
5
80 100
500
80
625
=
==
x
x .

So 5 is 6.25% of 80. The number 80 is called the base of the percent. Another
way to view this problem is to simply divide 5 by the base, 80, and then multiply
the result by 100 to get the percent.
If a quantity increases from 600 to 750, then the percent increase is found by
dividing the amount of increase, 150, by the base, 600, which is the first (or the
smaller) of the two given numbers, and then multiplying by 100:
150
600
100 25%




=()% .

If a quantity decreases from 500 to 400, then the percent decrease is found by
dividing the amount of decrease, 100, by the base, 500, which is the first (or the
larger) of the two given numbers, and then multiplying by 100:
10 0
500
10 0 20




=()% %.

Other ways to state these two results are “750 is 25 percent greater than 600”
and “400 is 20 percent less than 500.”
In general, for any positive numbers x and y, where x < y,
y is
y
x
x
-




()100 percent greater than x
x is
y
x
y
-




()100 percent less than y
Note that in each of these statements, the base of the percent is in
the denominator.
12


1.7 Ratio

The ratio of the number 9 to the number 21 can be expressed in several ways;
for example,
9 to 21
9:21
9
21

Since a ratio is in fact an implied division, it can be reduced to lowest terms.
Therefore, the ratio above could also be written:
3 to 7
3:7
3
7







1.8 Absolute Value

The absolute value of a number N, denoted by
N , is defined to be N if N
is positive or zero and –N if N is negative. For example,
1
2
1
2
00==,,
and
-= =26 26 26.(.)

Note that the absolute value of a number cannot be negative.
13


ARITHMETIC EXERCISES

(Answers on pages 17 and 18)

1. Evaluate:
(a) 15
– (6 – 4)(–2) (e) (–5)(–3) – 15
(b) (2
– 17) ÷ 5 (f) (–2)
4
(15 – 18)
4

(c) (60
÷ 12) – (–7 + 4) (g) (20 ÷ 5)
2
(–2 + 6)
3

(d) (3)
4
– (–2)
3
(h) (–85)(0) – (–17)(3)

2. Evaluate:
(a)
1
2
1
3
1
12
-+
(c)
7
8
4
5
2
-





(b)
3
4
1
7
2
5
+




-




(d)
3
8
27
32-











3. Evaluate:
(a) 12.837 + 1.65 – 0.9816 (c) (12.4)(3.67)
(b) 100.26 ÷ 1.2 (d) (0.087)(0.00021)

4. State for each of the following whether the answer is an even integer or
an odd integer.
(a) The sum of two even integers
(b) The sum of two odd integers
(c) The sum of an even integer and an odd integer
(d) The product of two even integers
(e) The product of two odd integers
(f) The product of an even integer and an odd integer

5. Which of the following integers are divisible by 8 ?
(a) 312 (b) 98 (c) 112 (d) 144

6. List all of the positive divisors of 372.

7. Which of the divisors found in #6 are prime numbers?

8. Which of the following integers are prime numbers?
19, 2, 49, 37, 51, 91, 1, 83, 29

9. Express 585 as a product of prime numbers.

14

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