## CHAPTER SUMMARY

### I. Divisibility

Divisibility

When two non-zero integers are multiplied, each integer is a factor of the product. The integer a is said to be divisible by the integer b if b is a factor of a. This means a can be divided by b with an integer result (meaning there is no remainder).

#### Factoring

By factoring a number, you break it down into as many factors as possible. When a question asks how many factors a number has, always remember to include:

• 1, because 1 is a factor of ALL integers.
• The number itself, because any integer is divisible by itself.

#### Prime Numbers

prime number is an integer greater than 1 whose only positive factors are 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

#### Prime Factorization

You can break a number down into its prime factors. You do this by finding factors using the divisibility rules.  Start with the smallest factors.
140 = 2 × 70 = 2 × 2 × 35 = 2 × 2 × 5 × 7

#### The Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor (divisor) for two numbers. GCF can be used to reduce fractions to simplest form.

How to get the GCF:

1. List the prime factors of each number.
2. See which prime factors are in both lists.
3. Multiply the factors both numbers share. If there are no common prime factors, then the GCF is 1.

#### Factors and Multiples

A multiple of an integer n is the product of n and any integer. To remember the difference between factors and multiples, use the mnemonic “Fewer Factors, More Multiples.”

#### Least Common Multiple (LCM)

common multiple of two or more integers is an integer that is a multiple of all of them. You can find the least common multiple by making a list of multiples for all the integers and identifying the smallest number that shows up on all the lists – this number will be the LCM.

LCM can be used to find the common denominator for adding or subtracting fractions.

### II. Fractions

Fractions

fraction is a number of the form ±\dfrac{\textit{a}}{\textit{b}}, where a and b are positive integers. The number a is called the numerator and is called the denominator.

#### Equivalent Fractions

A fraction that has a common factor in both the numerator and denominator is equal to the fraction with the common factor canceled.

\dfrac{6}{10} = \dfrac{3}{5}
(factor 2 in numerator and denominator of \dfrac{6}{10})

#### Simplifying Fractions

To simplify fractions, one method is to use the GCF (greatest common factor). Divide the numerator and denominator by the GCF to reduce the fraction. The GMAT always puts fractions in the lowest possible terms, so you should reduce any fractions to the simplest terms before going to the answer choices.

#### Mixed Numbers

Mixed numbers are numbers that are an integer plus a proper fraction.

The number 4\dfrac{\,2\,}{3} is the integer 4, plus the fraction \dfrac{\,2\,}{3}

#### Multiplying Fractions

How to multiply fractions:

1. Cancel out any common factors that appear in numerators and denominators.
2. Multiply all numerators to form one numerator and all denominators to form one denominator

#### Factors and Multiples

A multiple of an integer n is the product of n and any integer. To remember the difference between factors and multiples, use the mnemonic “Fewer Factors, More Multiples.”

#### Dividing Fractions

To divide fractions:

1. Change the divisor to its reciprocal, then multiply the fractions. (Remember that the divisor is the second number.)
2. The reciprocal of \dfrac{2}{3} is \dfrac{3}{2}.  The reciprocal of \dfrac{5}{4} is \dfrac{4}{5}.
3. When multiplying, cancel out any common factors that appear in both numerators and denominators.

#### Adding and Subtracting Fractions

• To add or subtract fractions that have the same denominator, add or subtract the numerators. Check to see that the answer is in simplest terms.
• To add or subtract fractions that have different denominators, use the least common denominator (LCD). The LCD is the least common multiple (LCM) of all the denominators.

#### Complex Fractions

complex fraction is a fraction that has a fraction in the numerator and/or denominator. In other words, it is a fraction divided by a fraction. To simplify, use the reciprocal of the divisor, then multiply.

#### Cross Multiplication

An easy way to compare fractions is cross multiplication. Multiply the numerator of one fraction by the denominator of the other fraction and compare the products.

### III. Decimals

Decimals

Decimals are numbers written using the base-ten place-value system. Each place value is 10 times the value to its right.

#### Rounding

For the specific digit:

• If the digit is less than 5, round down.
• If the digit is 5 or greater, round up.

#### Decimals and Fractions

Decimals can be used to write fractions. If the fraction has a denominator that is a multiple of ten, and the numerator does not end in 0, then the number of zeros in the denominator will be the number of places to the right of the decimal.

For other fractions, use the fraction bar as a division symbol.

#### Adding and Subtracting Decimals

To add or subtract decimals, use a vertical format and line up the decimals. You can add trailing zeros as placeholders at the end of a decimal to make the alignment easier.

#### Multiplying and Dividing Decimals

Multiply two decimals just like you would multiply two integers, then place the decimal point in the answer. The number of decimal places in the product is equal to the sum of number of the decimal places in the two factors.

To divide decimals, move the decimal point in the divisor to the right so it’s a whole number. Then move the decimal point in the dividend the same number of places.

### IV. Exponents

Exponents

Exponents express the multiplication of a number by itself. The exponent says how many times the number, called the base, is used as a factor. An expression that has a base and exponent is called a power.

#### Common Values

You should memorize these powers that appear frequently on the GMAT to save time on test day:

• Squares: up to 152 = 225
• Cubes: up to 63 = 216
• Base of 2: up to 210 = 1,024
• Base of 3: up to 35 = 243
• Base of 5: up to 54 = 625
• Base of 10: up to 104 = 10,000

#### Rules of Exponents

There are many rules when it comes to evaluating and manipulating exponents. Be sure to memorize these rules for the GMAT.

#### Negative Exponents

The reciprocal for a number with a positive exponent, \dfrac{1}{\textit{a}^{\displaystyle{\textit{n}}}} (n is a positive integer), is equivalent to the number with the negative exponentan.
\\[1ex]\textit{a}^{\displaystyle{-\textit{n}}} = \dfrac{1}{\textit{a}^{\displaystyle{\textit{n}}}}

#### Simplifying Exponential Expressions

Looking at the answers before starting to solve the problem is a good strategy for questions with exponents.  You may match the answer format by just simplifying or estimating, rather than calculating to get a value.

### V. Roots

Roots

A root is shown using a radical symbol \surd.  The number in front of the radical symbol is the degree of the root.  For square roots, the degree 2 is left off.

#### Common Values

As with exponents, you should memorize common roots for the GMAT. Luckily, since roots and exponents are connected, knowing the powers also tells you the value of the root. For any positive integer n, the nth root of a number x is r such that rn = x.

#### Fractional Exponents

When a base is a non-negative number, fractional exponents are another way to indicate a root.  When the exponent is a fraction, the numerator says what power to raise the base to, and the denominator says what root to take.

#### Roots and Negative Numbers

Not all roots are possible.  For example, \sqrt{-4} is impossible since no number raised to an even power is negative.  But odd roots of negative numbers are possible.

\sqrt[\displaystyle{3}]{-125} = -5 since (-5)3 = -125

In order to avoid double-value of even roots, the radical sign with even degree is a non-negative value only.

For example, \sqrt{4} = 2 \,\,(\sqrt{4} \mathrel{\char≠} -2),
\sqrt[\displaystyle{4}]{81} = 3 \,\,(\sqrt[\displaystyle{4}]{81} \mathrel{\char≠} -3).

#### Estimating Roots

Not all roots yield an integer. It is usually enough to just simplify expressions by factoring rather than multiplying to get a value.

162 = \sqrt{(2 × 81)} = 9\sqrt{2}

If the answer choices are numbers, you can calculate to get an estimated value. How exact the answer choices are will tell you how accurate an estimate you need to make.