Definition

Number 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 Types

GMAT deals with only Real Numbers: Integers, Fractions and Irrational Numbers.

INTEGERS

Definition
Integers are defined as: all negative natural numbers {...,−4,−3,−2,−1}, zero {0}, and
positive natural numbers {1,2,3,4,...}.

Note that integers do not include decimals or fractions - just whole numbers.

Even and Odd Numbers

An 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 n=2k, where k 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 n=2k+1, where k 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 NUMBERS
Fractions (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 2√ ("the square root of two") or the
number pi (π=~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 NUMBERS

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.

Multiplication:
positive * positive = positive
positive * negative = negative
negative * negative = positive

Division:
positive / positive = positive
positive / negative = negative
negative / negative = positive

Prime Numbers
A Prime number is a natural number with exactly two distinct natural number divisors: 1 and
itself. Otherwise a number is called acomposite number. Therefore, 1 is not a prime, since
it only has one divisor, namely 1. A number n>1 is prime if it cannot be written as a product
of two factors a and b, 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 6n−1 or 6n+1, because all other numbers are
divisible by 2 or 3.

• Any nonzero natural number n can be factored into primes, written as a product of primes
or powers of primes. Moreover, thisfactorization 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

unique prime factors a, b, and c can be expressed as n=ap∗bq∗cr, where p, q, and r are
one or more prime integers in a way which is unique. For instance integer n with three

Example: 4200=23∗3∗52∗7.
powers of a, b, and c, respectively and are ≥1.

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

number n can be done by trial division, that is to say dividing n by all integer numbers
smaller than n√, thereby checking whether n is a multiple of m≤n√.
Example: Verifying the primality of 161: 161−−−√ is little less than 13, from integers
from 2 to 13, 161 is divisible by 7, hence 161 is not prime.
Note that, it is only necessary to try dividing by prime numbers up to n√, since if n has any
divisors at all (besides 1 and n), then it must have a prime divisor.

• If n is a positive integer greater than 1, then there is always a prime

number p withn<p<2n.

Factors
A divisor of an integer n, also called a factor of n, is an integer which evenly
divides n without leaving a remainder. In general, it is said m is a factor of n, for non-zero
integers m and n, if there exists an integer k such that n=km.

• 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 a is a factor of b and a is a factor of c, then a is a factor of (b+c). In fact, a is a factor
of (mb+nc) for all integers m and n.

• If a is a factor of b and b is a factor of c, then a is a factor of c.

• If a is a factor of b and b is a factor of a, then a=b or a=−b.

• If a is a factor of bc, and gcd(a,b)=1, then a is a factor of c.

• If p is a prime number and p is a factor of ab then p is a factor of a or p is a factor of b.

Finding the Number of Factors of an Integer

First make prime factorization of an integer n=ap∗bq∗cr, where a, b, and c are prime
factors of n and p, q, and r are their powers.

The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). NOTE: this
will include 1 and n itself.

Example: Finding the number of all factors of 450: 450=21∗32∗52

Total number of factors of 450 including 1 and 450 itself

is (1+1)∗(2+1)∗(2+1)=2∗3∗3=18 factors.

Finding the Sum of the Factors of an Integer

First make prime factorization of an integer n=ap∗bq∗cr, where a, b, and c are prime
factors of n and p, q, and r are their powers.

The sum of factors of n will be expressed by the formula: (ap+1−1)∗(bq+1−1)∗(cr+1−1)

(a−1)∗(b−1)∗(c−1)

Example: Finding the sum of all factors of 450: 450=21∗32∗52

The sum of all factors of 450 is (21+1−1)∗(32+1−1)∗(52+1−1)
(2−1)∗(3−1)∗(5−1)=3∗26∗1241∗2∗4=1209

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 - LCM

The 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 Square
A 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 Rules
2 - 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.

Factorials
Factorial of a positive integer n, denoted by n!, is the product of all positive integers less
than or equal to n. For instance 5!=1∗2∗3∗4∗5.

• 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 n, can be determined with this formula:

n5+n52+n53+...+n5k, where k must be chosen such that 5k≤n.

It's easier if you look at an example:

How many zeros are in the end (after which no other digits follow) of 32!?
325+3252=6+1=7 (denominator must be less than 32, 52=25 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 p, in the n!.

The formula is:

np+np2+np3 ... till px≤n

What is the power of 2 in 25!?

252+254+258+2516=12+6+3+1=22

Finding the power of non-prime in n!:

How many powers of 900 are in 50!

Make the prime factorization of the number: 900=22∗32∗52, then find the powers of these
prime numbers in the n!.

Find the power of 2:

502+504+508+5016+5032=25+12+6+3+1=47

= 247

Find the power of 3:

503+509+5027=16+5+1=22

=322

Find the power of 5:

505+5025=10+2=12

=512

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

• Sum of n consecutive integers equals the mean multiplied by the number of terms, n.

average of the first and last terms), so the sum equals to −12∗6=−3.
Given consecutive integers {−3,−2,−1,0,1,2}, mean=−3+22=−12, (mean equals to the

• If n is odd, the sum of consecutive integers is always divisible by n. Given {9,10,11}, we

have n=3 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 {9,10,11,12},

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

• The product of n consecutive integers is always divisible by 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.

Evenly Spaced Set

Evenly 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 {9,13,17,21} is an example of evenly spaced set. Set of consecutive integers is
also an example of evenly spaced set.

• If the first term is a1 and the common difference of successive members is d, then
the nth term of the sequence is given by:
an=a1+d(n−1)

• In any evenly spaced set the arithmetic mean (average) is equal to the median and
can be calculated by the formula mean=median=a1+an2, where a1 is the first term
and an is the last term. Given the set {7,11,15,19}, mean=median=7+192=13.

Sum=a1+an2∗n, the mean multiplied by the number of terms. OR, Sum=2a1+d(n−1)2∗n

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

Sum of n first positive integers: 1+2+...+n=1+n2∗n

• Special cases:

Sum of n first positive odd numbers: a1+a2+...+an=1+3+...+an=n2, where an is the

last, nth term and given by: an=2n−1. Given n=5 first odd positive integers, then their
sum equals to 1+3+5+7+9=52=25.

Sum of n first positive even numbers: a1+a2+...+an=2+4+...+an=n(n+1), where an is

the last, nth term and given by: an=2n. Given n=4 first positive even integers, then their
sum equals to 2+4+6+8=4(4+1)=20.

• 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.

FRACTIONS

Definition
Fractional 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 (mn) and decimal
representation (a.bcd).

Fractional representation
Fractional 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, 97, 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. 13 is a proper fraction.

• Fractions that are greater than 1 are called improper fraction. Improper fraction can also
be written as a mixed number. 52 is improper fraction.

• An integer combined with a proper fraction is called mixed number. 435 is a mixed
number. This can also be written as an improper fraction: 235

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 114 to a mixed fraction.
Solution: Divide 114=2 with a remainder of 3. Write down the 2 and then write down the
remainder 3 above the denominator 4, like this: 234

• 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 325 to an improper fraction.
Solution: Multiply the whole number by the denominator: 3∗5=15. Add the numerator to
that: 15+2=17. Then write that down above the denominator, like this: 175

Reciprocal
Reciprocal for a number x, denoted by 1x or x−1, is a number which when multiplied
by x yields 1. The reciprocal of a fraction ab is ba. To get the reciprocal of a number, divide
1 by the number. For example reciprocal of 3 is 13, reciprocal of 56 is 65.

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: 37+23=921+1421=2321

multiply: 35∗12=310.
Example #2: Given 352, take the reciprocal of 2. The reciprocal is 12. Now

Decimal Representation
The 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 ab (meaning that fraction is already reduced to its lowest term) can be
expressed as terminating decimal if and only b(denominator) is of the form 2n5m,
where m and n are non-negative integers. For example: 7250 is a terminating
decimal 0.028, as 250(denominator) equals to 2∗53. Fraction 330 is also a terminating
decimal, as 330=110 and denominator 10=2∗5.

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 0.56 to a fraction.

1: Total number after decimal point is 2.
2 and 3: 56100.
4: Reducing it to lowest terms: 56100=1425

• 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 0.393939... to a fraction.

1: The recurring number is 39.
2: 3999, the number 39 is of length 2 so we have added two nines.
3: Reducing it to lowest terms: 3999=1333.

• 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 0.2512(12) 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.

Rounding
Rounding 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 Proportions

Given that ab=cd, 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.

ba=dc - invertendo

ac=bd - alternendo

a+bb=c+dd - componendo

a−bb=c−dd - dividendo

a+ba−b=c+dc−d - componendo & dividendo

EXPONENTS
Exponents are a "shortcut" method of showing a number that was multiplied by itself several
times. For instance, number a multiplied ntimes can be written as an, where a represents
the base, the number that is multiplied by itself n times and n represents the exponent. The
exponent indicates how many times to multiple the base, a, by itself.

Exponents one and zero:

a0=1 Any nonzero number to the power of 0 is 1.
For example: 50=1 and (−3)0=1
• Note: the case of 0^0 is not tested on the GMAT.

a1=a Any number to the power 1 is itself.

Powers of zero:
If the exponent is positive, the power of zero is zero: 0n=0, where n>0.

If the exponent is negative, the power of zero (0n, where n<0) is undefined, because
division by zero is implied.

Powers of one:
1n=1 The integer powers of one are one.

Negative powers:
a−n=1an

Powers of minus one:

If n is an even integer, then (−1)n=1.

If n is an odd integer, then (−1)n=−1.

Operations involving the same exponents:

an∗bn=(ab)n
Keep the exponent, multiply or divide the bases

anbn=(ab)n

(am)n=amn

and not (am)n

Operations involving the same bases:

an∗am=an+m
Keep the base, add or subtract the exponent (add for multiplication, subtract for division)

anam=an−m

Fraction as power:
a1n=a√n

amn=am−−−√n

Exponential Equations:
When solving equations with even exponents, we must consider both positive and negative
possibilities for the solutions.

For instance a2=25, the two possible solutions are 5 and −5.

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

For instance for a3=8, solution is a=2 and for a3=−8, solution is a=−2.

Exponents and divisibility:

an−bn is ALWAYS divisible by a−b.
an−bn is divisible by a+b if n is even.

an+bn is divisible by a+b if n is odd, and not divisible by a+b if n is even.

LAST DIGIT OF A PRODUCT

Last n digits of a product of integers are last n digits of the product of last n 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=540*104=40*4=160=60

Example: The last digit of 85945*89*58307=5*9*7=45*7=35=5?

LAST DIGIT OF A POWER

Determining the last digit of (xyz)n:

1. Last digit of (xyz)n is the same as that of zn;

2. Determine the cyclicity number c of z;
3. Find the remainder r when n divided by the cyclisity;
4. When r>0, then last digit of (xyz)n is the same as that of zr and when r=0, then last digit
of (xyz)n is the same as that of zc, where c 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. (xy4)n) have a cyclisity of 2. When n is odd (xy4)n will end
with 4 and when n is even (xy4)n will end with 6.
• Integers ending with 9 (eg. (xy9)n) have a cyclisity of 2. When n is odd (xy9)n will end
with 9 and when n is even (xy9)n will end with 1.

Example: What is the last digit of 12739?

Solution: Last digit of 12739 is the same as that of 739. Now we should determine the
cyclisity of 7:

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 12739 is the same
as that of the last digit of 739, is the same as that of the last digit of 73, which is 3.

ROOTS
Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16
and square root of 16=4.

General rules:
• x√y√=xy−−√ and x√y√=xy−−√.

• (x√)n=xn−−√

• x1n=x√n

• xnm=xn−−√m

• a√+b√≠a+b−−−−√

• x2−−√=|x|, when x≤0, then x2−−√=−x and when x≥0, then x2−−√=x

• When the GMAT provides the square root sign for an even root, such as x√ or x√4, then
the only accepted answer is the positive root.

That is, 25−−√=5, NOT +5 or -5. In contrast, the equation x2=25 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, 125−−
−√3=5 and −64−−−−√3=−4.

• For GMAT it's good to memorize following values:

2√≈1.41
3√≈1.73
5√≈2.24
6√≈2.45
7√≈2.65
8√≈2.83
10−−√≈3.16

PERCENTS
A percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per
hundred"). It is often denoted using the percent sign, "%", or the abbreviation "pct". Since a
percent is an amount per 100, percents can be represented as fractions with a denominator
of 100. For example, 25% means 25 per 100, 25/100 and 350% means 350 per 100,
350/100.

• A percent can be represented as a decimal. The following relationship characterizes how

percents and decimals interact. Percent Form / 100 = Decimal Form

For example: What is 2% represented as a decimal?

Percent Form / 100 = Decimal Form: 2%/100=0.02

• Percent change

General formula for percent increase or decrease, (percent change):

Percent=ChangeOriginal∗100

Example: A company received $2 million in royalties on the first $10 million in sales and
then $8 million in royalties on the next $100 million in sales. By what percent did the ratio of
royalties to sales decrease from the first $10 million in sales to the next $100 million in
sales?

Percent=ChangeOriginal∗100=
Solution: Percent decrease can be calculated by the formula above:

=210−8100210∗100=60, so the royalties decreased by 60%.

• Simple Interest
Simple interest = principal * interest rate * time
Example: If $15,000 is invested at 10% simple annual interest, how much interest is earned
after 9 months?
Solution: $15,000*0.1*9/12 = $1125

• Compound Interest
Balance(final)=principal∗(1+interestC)time∗C, where C = the number of times
compounded annually. If C=1, meaning that interest is compounded once a year, then the
formula will be: Balance(final)=principal∗(1+interest)time, where time is number of
years.
Example: If $20,000 is invested at 12% annual interest, compounded quarterly, what is the
balance after 2 year?
Solution: Balance=20,000∗(1+0.124)2∗4=20,000∗(1.03)8=$25,335.4

ORDER OF OPERATIONS - PEMDAS

Perform the operations inside a Parenthesis first (absolute value signs also fall into this
category), then Exponents, then Multiplication andDivision, 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 32! means (32)!=9! while 25!=2120; 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 (am)n