NUMBER THEORYDefinitionNumber Theory is concerned with the properties of numbers in general, and in particular integers.
As this is a huge issue we decided to divide it into smaller topics. Below is the list of Number Theory topics.
GMAT Number TypesGMAT is dealing only with
Real Numbers: Integers, Fractions and Irrational Numbers.
INTEGERSDefinitionIntegers are defined as: all negative natural numbers

, zero

, and positive natural numbers

.
Note that integers do not include decimals or fractions - just whole numbers.Even and Odd NumbersAn even number is an
integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder.
An even number is an integer of the form

, where

is an integer.
An odd number is an
integer that is not evenly divisible by 2.
An odd number is an integer of the form

, where

is an integer.
Zero is an even number.Addition / Subtraction:even +/- even = even;
even +/- odd = odd;
odd +/- odd = even.
Multiplication:even * even = even;
even * odd = even;
odd * odd = odd.
Division of two integers can result into an even/odd integer or a fraction.
IRRATIONAL NUMBERSFractions (also known as rational numbers) can be written as
terminating (ending) or
repeating
decimals (such as 0.5, 0.76, or 0.333333....). On the other hand, all
those numbers that can be written as non-terminating, non-repeating
decimals are non-rational, so they are called the "irrationals".
Examples would be

("the square root of two") or the number pi (

~3.14159..., from geometry). The rationals and the irrationals are two totally separate number types: there is no overlap.
Putting these two major classifications, the rationals and the irrationals, together in one set gives you the "real" numbers.
POSITIVE AND NEGATIVE NUMBERSA 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.Multiplication: positive * positive = positive
positive * negative = negative
negative * negative = positive
Division:positive / positive = positive
positive / negative = negative
negative / negative = positive
Prime NumbersA
Prime number is a natural number with exactly two distinct natural
number divisors: 1 and itself. Otherwise a number is called a
composite number. Therefore,
1 is not a prime, since it only has one divisor, namely 1. A number

is prime if it cannot be written as a product of two factors

and

, both of which are greater than 1: n = ab.
• The first twenty-six prime numbers 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, 97, 101
• Note: only positive numbers can be primes.• There are infinitely many prime numbers.
• The
only even prime number is 2, since any larger even number is divisible by 2. Also
2 is the smallest prime.
•
All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly,
all prime numbers above 3 are of the form
or 
, because all other numbers are divisible by 2 or 3.
• Any nonzero natural number

can be factored into primes, written as a product of primes or powers of primes. Moreover, this
factorization is unique except for a possible reordering of the factors.
•
Prime factorization:
every positive integer greater than 1 can be written as a product of
one or more prime integers in a way which is unique. For instance
integer

with three unique prime factors

,

, and

can be expressed as

, where

,

, and

are powers of

,

, and

, respectively and are

.
Example: 
.
•
Verifying the primality (checking whether the number is a prime) of a given number

can be done by trial division, that is to say dividing

by all integer numbers smaller than

, thereby checking whether

is a multiple of

.
Example: Verifying the primality of

:

is little less than

, from integers from

to

,

is divisible by

, hence

is not prime.
• If

is a positive integer greater than 1, then there is always a prime number

with

.
FactorsA divisor of an
integer 
, also called a factor of

, is an
integer which evenly divides

without leaving a remainder. In general, it is said

is a factor of

, for non-zero integers

and

, if there exists an integer

such that

.
• 1 (and -1) are divisors of every integer.
• Every integer is a divisor of itself.
• Every integer is a divisor of 0, except, by convention, 0 itself.
• Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.
• A positive divisor of n which is different from n is called a
proper divisor.
•
An integer n > 1 whose only proper divisor is 1 is called a prime
number. Equivalently, one would say that a prime number is one which has
exactly two factors: 1 and itself.
• Any positive divisor of n is a product of prime divisors of n raised to some power.
• If a number equals the sum of its proper divisors, it is said to be a
perfect number.
Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.
There are some elementary rules:
• If

is a factor of

and

is a factor of

, then

is a factor of

. In fact,

is a factor of

for all integers

and

.
• If

is a factor of

and

is a factor of

, then

is a factor of

.
• If

is a factor of

and

is a factor of

, then

or

.
• If

is a factor of

, and

, then a is a factor of

.
• If

is a prime number and

is a factor of

then

is a factor of
or 
is a factor of

.
Finding the Number of Factors of an IntegerFirst make prime factorization of an integer

, where

,

, and

are prime factors of

and

,

, and

are their powers.
The number of factors of

will be expressed by the formula

.
NOTE: this will include 1 and n itself.
Example: Finding the number of all factors of 450:

Total number of factors of 450 including 1 and 450 itself is

factors.
Finding the Sum of the Factors of an IntegerFirst make prime factorization of an integer

, where

,

, and

are prime factors of

and

,

, and

are their powers.
The sum of factors of

will be expressed by the formula:
Example: Finding the sum of all factors of 450:

The sum of all factors of 450 is

Greatest Common Factor (Divisior) - GCF (GCD)The
greatest common divisor (gcd), also known as the greatest common factor
(gcf), or highest common factor (hcf), of two or more non-zero
integers, is the largest positive integer that divides the numbers
without a remainder.
To find the GCF, you will need to do
prime-factorization. Then, multiply the common factors (pick the lowest
power of the common factors).
• Every common divisor of a and b is a divisor of gcd(a, b).
• a*b=gcd(a, b)*lcm(a, b)
Lowest Common Multiple - LCMThe
lowest common multiple or lowest common multiple (lcm) or smallest
common multiple of two integers a and b is the smallest positive integer
that is a multiple both of a and of b. Since it is a multiple, it can
be divided by a and b without a remainder. If either a or b is 0, so
that there is no such positive integer, then lcm(a, b) is defined to be
zero.
To find the LCM, you will need to do prime-factorization.
Then multiply all the factors (pick the highest power of the common
factors).
Perfect SquareA
perfect square, is an integer that can be written as the square of some
other integer. For example 16=4^2, is an perfect square.
There are some tips about the perfect square:
• The number of distinct factors of a perfect square is ALWAYS ODD.
• The sum of distinct factors of a perfect square is ALWAYS ODD.
• A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
• Perfect square always has even number of powers of prime factors.
Divisibility Rules2 - If the last digit is even, the number is divisible by 2.
3 - If the sum of the digits is divisible by 3, the number is also.
4 - If the last two digits form a number divisible by 4, the number is also.
5 - If the last digit is a 5 or a 0, the number is divisible by 5.
6 - If the number is divisible by both 3 and 2, it is also divisible by 6.
7
- Take the last digit, double it, and subtract it from the rest of the
number, if the answer is divisible by 7 (including 0), then the number
is divisible by 7.
8 - If the last three digits of a number are divisible by 8, then so is the whole number.
9 - If the sum of the digits is divisible by 9, so is the number.
10 - If the number ends in 0, it is divisible by 10.
11
- If you sum every second digit and then subtract all other digits and
the answer is: 0, or is divisible by 11, then the number is divisible by
11.
Example: to see whether
9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then
subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by
11, hence 9,488,699 is divisible by 11.
12 - If the number is divisible by both 3 and 4, it is also divisible by 12.
25 - Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25.
FactorialsFactorial of a
positive integer 
, denoted by

, is the product of all positive integers less than or equal to n. For instance

.
• Note: 0!=1.• Note: factorial of negative numbers is undefined.Trailing zeros:Trailing
zeros are a sequence of 0's in the decimal representation (or more
generally, in any positional representation) of a number, after which no
other digits follow.
125000 has 3 trailing zeros;
The number of trailing zeros in the decimal representation of
n!, the factorial of a non-negative integer

, can be determined with this formula:

, where k must be chosen such that

.
It's easier if you look at an example:
How many zeros are in the end (after which no other digits follow) of

?

(denominator must be less than 32,

is less)
Hence, there are 7 zeros in the end of 32!
The
formula actually counts the number of factors 5 in n!, but since there
are at least as many factors 2, this is equivalent to the number of
factors 10, each of which gives one more trailing zero.
Finding the number of powers of a prime number
, in the
.The formula is:

... till

What is the power of 2 in 25!?
Finding the power of non-prime in n!:How many powers of 900 are in 50!
Make the prime factorization of the number:

, then find the powers of these prime numbers in the n!.
Find the power of 2:


=

Find the power of 3:

=

Find the power of 5:

=

We
need all the prime {2,3,5} to be represented twice in 900, 5 can
provide us with only 6 pairs, thus there is 900 in the power of 6 in
50!.
Consecutive IntegersConsecutive
integers are integers that follow one another, without skipping any
integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers.
• Sum of

consecutive integers equals the mean multiplied by the number of terms,

. Given consecutive integers

,

, (mean equals to the average of the first and last terms), so the sum equals to

.
• If n is odd, the sum of consecutive integers is always divisible by n. Given

, we have

consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.
• If n is even, the sum of consecutive integers is never divisible by n. Given

, we have

consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.
• The product of

consecutive integers is always divisible by

.
Given

consecutive integers:

. The product of 3*4*5*6 is 360, which is divisible by 4!=24.
Evenly Spaced SetEvenly
spaced set or an arithmetic progression is a sequence of numbers such
that the difference of any two successive members of the sequence is a
constant. The set of integers

is an example of evenly spaced set. Set of consecutive integers is also an example of evenly spaced set.
• If the first term is

and the common difference of successive members is

, then the

term of the sequence is given by:

• In any evenly spaced set the arithmetic
mean (average) is equal to the median and can be calculated by the formula

, where

is the first term and

is the last term. Given the set

,

.
• The sum of the elements in any evenly spaced set is given by:

, the mean multiplied by the number of terms. OR,
• Special cases:Sum of n first integers:

Sum of n first odd numbers:

, where

is the last,

term and given by:

. Given

first odd integers, then their sum equals to

.
Sum of n first positive even numbers:


, where

is the last,

term and given by:

. Given

first positive even integers, then their sum equals to

.
•
If the evenly spaced set contains odd number of elements, the mean is
the middle term, so the sum is middle term multiplied by number of
terms. There are five terms in the set {1, 7, 13, 19, 25}, middle term
is 13, so the sum is 13*5 =65.
FRACTIONSDefinitionFractional
numbers are ratios (divisions) of integers. In other words, a fraction
is formed by dividing one integer by another integer. Set of Fraction is
a subset of the set of Rational Numbers.
Fraction can be expressed in two forms
fractional representation 
and
decimal representation 
.
Fractional representationFractional
representation is a way to express numbers that fall in between
integers (note that integers can also be expressed in fractional form). A
fraction expresses a part-to-whole relationship in terms of a numerator
(the part) and a denominator (the whole).
• The number on top of the fraction is called
numerator or
nominator. The number on bottom of the fraction is called
denominator. In the fraction,

, 9 is the numerator and 7 is denominator.
• Fractions that have a value between 0 and 1 are called
proper fraction. The numerator is always smaller than the denominator.

is a proper fraction.
• Fractions that are greater than 1 are called
improper fraction. Improper fraction can also be written as a mixed number.

is improper fraction.
• An integer combined with a proper fraction is called
mixed number.

is a mixed number. This can also be written as an improper fraction:
Converting Improper Fractions• Converting Improper Fractions to Mixed Fractions:
1. Divide the numerator by the denominator
2. Write down the whole number answer
3. Then write down any remainder above the denominator
Example #1: Convert

to a mixed fraction.
Solution: Divide

with a remainder of

. Write down the

and then write down the remainder

above the denominator

, like this:

• Converting Mixed Fractions to Improper Fractions:
1. Multiply the whole number part by the fraction's denominator
2. Add that to the numerator
3. Then write the result on top of the denominator
Example #2: Convert

to an improper fraction.
Solution: Multiply the whole number by the denominator:

. Add the numerator to that:

. Then write that down above the denominator, like this:
ReciprocalReciprocal for a number

, denoted by

or

, is a number which when multiplied by

yields

. The reciprocal of a fraction

is

. To get the reciprocal of a number, divide 1 by the number. For example reciprocal of

is

, reciprocal of

is

.
Operation on Fractions•
Adding/Subtracting fractions:To add/subtract fractions with the same denominator, add the numerators and place that sum over the common denominator.
To
add/subtract fractions with the different denominator, find the Least
Common Denominator (LCD) of the fractions, rename the fractions to have
the LCD and add/subtract the numerators of the fractions
•
Multiplying fractions: To multiply fractions just place the product of the numerators over the product of the denominators.
•
Dividing fractions: Change the divisor into its reciprocal and then multiply.
Example #1:
Example #2: Given

, take the reciprocal of

. The reciprocal is

. Now multiply:

.
Decimal RepresentationThe decimals has ten as its base. Decimals can be
terminating (ending) (such as 0.78, 0.2) or
repeating (recuring) decimals (such as 0.333333....).
Reduced fraction

(meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal
if and only 
(denominator) is of the form

, where

and

are non-negative integers. For example:

is a terminating decimal

, as

(denominator) equals to

. Fraction

is also a terminating decimal, as

and denominator

.
Converting Decimals to Fractions• To convert a terminating decimal to fraction:1. Calculate the total numbers after decimal point
2. Remove the decimal point from the number
3. Put 1 under the denominator and annex it with "0" as many as the total in step 1
4. Reduce the fraction to its lowest terms
Example: Convert

to a fraction.
1: Total number after decimal point is 2.
2 and 3:

.
4: Reducing it to lowest terms:
• To convert a recurring decimal to fraction:1. Separate the recurring number from the decimal fraction
2. Annex denominator with "9" as many times as the length of the recurring number
3. Reduce the fraction to its lowest terms
Example #1: Convert

to a fraction.
1: The recurring number is

.
2:

, the number

is of length

so we have added two nines.
3: Reducing it to lowest terms:

.
• To convert a mixed-recurring decimal to fraction:1. Write down the number consisting with non-repeating digits and repeating digits.
2. Subtract non-repeating number from above.
3.
Divide 1-2 by the number with 9's and 0's: for every repeating digit
write down a 9, and for every non-repeating digit write down a zero
after 9's.
Example #2: Convert

to a fraction.
1. The number consisting with non-repeating digits and repeating digits is 2512;
2. Subtract 25 (non-repeating number) from above: 2512-25=2487;
3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300.
RoundingRounding
is simplifying a number to a certain place value. To round the decimal
drop the extra decimal places, and if the first dropped digit is 5 or
greater, round up the last digit that you keep. If the first dropped
digit is 4 or smaller, round down (keep the same) the last digit that
you keep.
Example:5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.
Ratios and ProportionsGiven that

,
where a, b, c and d are non-zero real numbers, we can deduce other
proportions by simple Algebra. These results are often referred to by
the names mentioned along each of the properties obtained.

-
invertendo
-
alternendo
-
componendo
-
dividendo
-
componendo & dividendoEXPONENTSExponents are a "shortcut" method of showing a number that was multiplied by itself several times. For instance, number

multiplied

times can be written as

, where

represents the base, the number that is multiplied by itself

times and

represents the exponent. The exponent indicates how many times to multiple the base,

, by itself.
Exponents one and zero:
Any nonzero number to the power of 0 is 1.
For example:

and
• Note: the case of 0^0 is not tested on the GMAT.
Any number to the power 1 is itself.
Powers of zero:If the exponent is positive, the power of zero is zero:

, where

.
If the exponent is negative, the power of zero (

, where

) is undefined, because division by zero is implied.
Powers of one:
The integer powers of one are one.
Negative powers:
Powers of minus one:If n is an even integer, then

.
If n is an odd integer, then

.
Operations involving the same exponents:Keep the exponent, multiply or divide the bases




and not
Operations involving the same bases:Keep the base, add or subtract the exponent (add for multiplication, subtract for division)

Fraction as power:![a^{\frac{1}{n}}=\sqrt[n]{a}](math_nt_mimetex_062.gif)
Exponential Equations:When solving equations with
even exponents, we must consider both positive and negative possibilities for the solutions.
For instance

, the two possible solutions are

and

.
When solving equations with
odd exponents, we'll have only one solution.
For instance for

, solution is

and for

, solution is

.
Exponents and divisibility:
is ALWAYS divisible by

.

is divisible by

if

is even.

is divisible by

if

is odd, and not divisible by a+b if n is even.
LAST DIGIT OF A PRODUCTLast

digits of a product of integers are last

digits of the product of last

digits of these integers.
For instance last 2 digits of 845*9512*408*613 would be the last 2 digits of 45*12*8*13=5
40*1
04=40*4=1
60=60
Example: The last digit of 8594
5*8
9*5830
7=5*9*7=4
5*7=3
5=5?
LAST DIGIT OF A POWERDetermining the last digit of

:
1. Last digit of

is the same as that of

;
2. Determine the cyclicity number

of

;
3. Find the remainder

when

divided by the cyclisity;
4. When

, then last digit of

is the same as that of

and when

, then last digit of

is the same as that of

, where

is the cyclisity number.
• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.
• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.
• Integers ending with 4 (eg.

) have a cyclisity of 2. When n is odd

will end with 4 and when n is even

will end with 6.
• Integers ending with 9 (eg.

) have a cyclisity of 2. When n is odd

will end with 9 and when n is even

will end with 1.
Example: What is the last digit of

?
Solution: Last digit of

is the same as that of

. Now we should determine the cyclisity of

:
1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...
So, the cyclisity of 7 is 4.
Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of

is the same as that of the last digit of

, is the same as that of the last digit of

, which is

.
ROOTSRoots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.
General rules:
•

and

.
•

•
![x^{\frac{1}{n}}=\sqrt[n]{x}](math_nt_mimetex_049.gif)
•
![x^{\frac{n}{m}}=\sqrt[m]{x^n}](math_nt_mimetex_034.gif)
•

•

, when

, then

and when

, then

• When the GMAT provides the square root sign for an even root, such as

or
![\sqrt[4]{x}](math_nt_mimetex_241.gif)
, then the only accepted answer is the positive root.
That is,

, NOT +5 or -5. In contrast, the equation

has TWO solutions, +5 and -5.
Even roots have only a positive value on the GMAT.• Odd roots will have the same sign as the base of the root. For example,
![\sqrt[3]{125} =5](math_nt_mimetex_064.gif)
and
![\sqrt[3]{-64} =-4](math_nt_mimetex_164.gif)
.
• For GMAT it's good to memorize following values:






ORDER OF OPERATIONS - PEMDASPerform the operations inside a
Parenthesis first (absolute value signs also fall into this category), then
Exponents, then
Multiplication and
Division, from left to right, then
Addition and
Subtraction, from left to right - PEMDAS.
Special cases:
•
An exclamation mark indicates that one should compute the factorial of
the term immediately to its left, before computing any of the
lower-precedence operations, unless grouping symbols dictate otherwise.
But

means

while

; a factorial in an exponent applies to the exponent, while a factorial not in the exponent applies to the entire power.
• If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:

and not
