GMAT: Rules of Divisibility

Absolute value of a difference

Ansolute value of x-y can be interpreted as the distance between x and y. For example, rephrase the absolute value of x-3<4 as "the distance between x and 3 is less than 4"

Odds / evens with multiple variables

Approach by testing different odds/evens for each variable

Polygons and Area

Area refers to the space inside the polygon measured in square units

Add/Subtract two multiples

results is a multiple Multiple of 3 + Multiple of 3 = Multiple of 3

Powers and Roots

(0.04)^3 = 0.000064 --> (0.04)^3 has 2 places so 2 places x power of 3 = 6 places in the final answer (0.000000008) ^(1/3) = 0.002 --> (0.000000008)^(1/3) has 9 places so 9 places /power of 3 = 3 places in the final answer

Sum of Consecutive Integers

(1) Average the first and last term to find the middle of the set (2) Count the number of terms (3) Multiply the middle term by the number of terms to find the set

Properties of Evenly Spaced Sets

(1) Mean and the median are equal, (2) mean and median of the set are equal to the average of the FIRST and LAST terms, (3)the sum of the elements in the set equals the arithmetic mean x number of items in set

Property of GCF and LCM

(GCF of m and n) x (LCM of m and n) = m x n

An exponent of 0

Any nonzero base raised to the power of zero is equal to 1

Never split the denominator

The numerator may be split, but never split the denominator.

Sums of Consecutive Integers and Divisibility

The sum of n consecutive integers is divisible by n if n is odd, but it is NOT divisible by n if n is even

Beware of even exponents

They hide the original sign of the base. Any base raised to an even power will result in a positive answer

Area of Rectangle

Length x Width

Divisible by 2 if

Even

Simplifying Roots with Prime Factorization

1) Factor the number under the radical sign into primes 2) Pull out any pair of matching primes from under the randical sign, place one of them outside the root 3) Consolidate the expression

Estimating Decimal Equivalents

1) Make the denominator the nearest factor of 100 or another power of 10. 2) Change the numerator or denominator to make the fraction simplify easily. 3) Make percent adjustments by seeing how much you approximately changed the denominator and applying that percent to the final answer.

Exponents Strategy

1) Simplify or factor any additive or subtractive terms 2) Break every non-prime base down into prime factors 3) Distribute the exponents to every prime factor 4) Combine the exponents for each prime factor and simplify

Evenly Spaced Sets

All sets of consecutive integers are sets of consecutive multiples. All sets of consecutive multiples are evenly spaced sets. All evenly spaced sets are fully defined if the 3 parameters are known: (1) first and last numbers in the set, (2) the increment, (3) the number of items in the set

Rhombus

All sides are equal. Opposite angles are equal.

Counting Integers for consecutive integers (how many integers from x to y?)

Add one before you are done For example, how many integers between 6 and 10? Count 6, 7, 8, 9, 10 or just subtract 10-6 + 1 (Last - First + 1)

Square

All angles are 90 degrees and all sides are equal

Rectangle

All angles are 90 degrees and opposite sides are equal

Factors of perfect squares

All perfect squares have an odd number of total factors. Vice versa so if a integer has an odd number of total factors it must be a perfect square

Numerator and denominator rules for positive numbers

As numerator goes up, the fraction increases. As denominator goes up, the fraction decreases. Adding the same number to both numerator and denominator brings the fraction closer to 1 regardless of the fraction's value. If the fraction is originally smaller than 1, the fraction increases in value as it approaches 1. if the fraction is originally larger than 1, the fraction decreases in value as it approaches 1.

Area of Parallelogram

Base x Height

Factorials and Divisibility

Because it is a product of all the integers from 1 to N, any factorial N! must be divisible by all integers from 1 to N. N! is a multiple of integers from all the integers from 1 to N

Successive Percents

Best to solve by choosing real numbers and seeing what happens-- 100 is usually the easiest number

Representing Evens and Odds Algebraically

Even numbers can be written as 2n Odd numbers can be written as 2n+1 or 2n-1

Divisible by 8 if

Integer is divisible by 2 three times or if the last three digits are divisible by 8

Strategy for solving data sufficiency- type of problem

Is it a value question or a yes/no question? Never turn a yes/no question into a value question

Add/Subtract Decimals

Line up the decimal points!

Powers of 10- divide by positive power of 10

Move decimal backward (left) to make the positive number smaller. For example: 4169.2 / 10^2 = 41.692 (move backwards 2 spaces) 89.507 / 10 = 8.9507 (move backwards 1 space)

Powers of 10- multiply by positive power of 10

Move decimal forward (right) to make the positive number larger. For example: 3.9742 x 10^3 = 3974.2 (move decimal forward 3 spaces) 89.507 x 10 = 895.07 (move decimal forward 1 space) Add zeros if needed: 2.57 x 10^6 = 2,570,000 14.29 / 10^5 = 0.0001429

Chemical Mixtures

Set up a mixture chart with the substance labels in rows and "original," "change" and "new" in the columns. This way you can keep track of various components and their changes.

Add/Subtract Odds & Evens

ODD +_ EVEN = ODD ODD + _ ODD = EVEN EVEN + _ EVEN = EVEN

Multiply/Divide Odds & Evens

ODD x ODD = ODD EVEN x EVEN = EVEN (and divisible by 4) ODD x EVEN = EVEN

Trapezoid

One pair of opposite sides is parallel

Teminating Decimals

Only have prime factors of 2's and 5's. If there are other prime factors, it is not a terminating decimal.

Parallelogram

Opposite sides and opposite angles are equal

Percents as Fractions

Part/Whole = Percent/100 Fill in the table, set up as a proportion, cancel cross-multiply and solve

Polygons and perimeter

Perimeter is the distance around a polygon and equals the sum of all sides

Negative Exponents

Raising a number to a negative exponent is the same as raising the number's reciprocal to the equivalent positive exponent

Stategies to solve data sufficiency - rephrase

Rephrase: take the given information and reduce to its simplest form then focus on how the piece of info relates to the question

Data Sufficiency strategy

Rephrasing: You should be able to rephrase the equation to have one equation with one variable. If you are unable to do this, the stem is not sufficient.

Factor Counting

Solve factor counting problems by writing the prime factorization in exponential form, adding 1 to all of the exponents and multiplying

Polygons and Interior Angles

Sum of Interior Angles of a polygon = (n-2) x 180 Triangle has 3 sides and 180 degrees Quadrilateral has 4 sides and 360 degrees Pentagon has 5 sides and 540 degrees Hexagon has 6 sides and 720 degrees Another way to find the sum is to divide the polygon into triangles

Divisible by 3 if

Sum of the integer's digits are divisble by 3

Property of GCF and LCM

The GCF of m and n cannot be larger than the difference between m and n. For exmaple, assume the GCF of m and n is 12. Thus m and n are both multiples of 12. Consecutive multiples of 12 are 12 units apart on the number line and therefore cannot be less than 12 units apart

Percent problems

The key is to make them concrete by picking real numbers with which to work

Benchmark Values: 10%

To find 10% of any number, just move the decimal point to the left one place

Sum of two primes

Unless one of the numbers is 2, it will result in an even number

Unspecified Number Amounts

Use Smart numbers. To make computation easier, choose numbers equal to common multiples of the denominators of the fraction in the problem.

Percent Increase and decrease

Use the percent table however adjust it for change instead of part. Change/Original = Percent/100 Also can do so with following equations: ORIGINAL x (1+% increase/100) = NEW ORIGINAL x (1-% decrease/100) = NEW

Heavy Division Shortcut

Used for large numbers. Example: What is 1,530,794 / (31.49 x 10^4) to the nearest whole number? Step 1: Set up the division problem in fraction form Step 2: Rewrite the problem, eliminating powers of 10: 1,530,794 / 314,900 Step 3: The goal is to get a single digit to the left of the decimal in the denominator. Just remember whatever you do to the denominator, to do to the numerator. 1,530,794 / 314,900 = 15.30794 / 3.14900 Now you have the single digit 3 to the left of the decimal in the denominator Step 4: Focus only on the whole number parts and solve. 15.30794 / 3.14900 is approx 15/3 which = 5

Disguised + or - questions

Whenever you see >0 or <0, think Positives & Negatives

Quadrilaterals

four sided shapes trapezoids, parallelograms and special parallelograms such as rhombuses, rectangles and squares

Divisible by 10 if

integer ends in 0

Powers of 10-negative powers

Negative powers reverse the process. Ex: 6782.01 x 10^-3 = 6.78201 (Moving the decimal forward by negative 3 spaces means moving it backward by 3 spaces) 53.0447 / 10^-2 = 5304.47 (Moving the decimal backward by negative 2 spaces means moving it forward by 2 spaces)

No addition or subtraction shortcuts with fractions

1)find a common denominator 2) change each fraction so that it is expressed using this common denominator 3) add up the numerators only

When to Simplify Eponential Expressions

(1) You can only simplify expressions that are linked by multiplication or division (not addition or subtraction) (2) You can simplify expressions linked by multiplication or division if they have either a base or an exponent in common

Facts about sums and averages of evenly spaced sets

(1) the average of an ODD number of consecutive integers will always be an integer (2) the average of an EVEN number of consecutive integers will never be an integer

Area of a Triangle

(Base x Height) / 2

Counting Integers for consecutive multiples (how many integers from x to y?)

(Last - First) / Increment + 1

When to try plugging in numbers

1) variables in the answer choices 2) percents in the answer choices (when they are percents of some unspecified amount) 3)Fractions or ratios in teh answer choices (when they are fractional parts or ratios of unspecified amounts)

Comparing Fractions

Cross Multiply Cross multiply the fractions and put each answer by the corresponding numerator For example: 7/9 vs. 4/5 (7 x 5) = 35 (4 x 9) = 36 Put 35 next to corresponding 7/9 and 36 next to corresponding 4/5. Since 36 is larger than 35, 4/5 > 7/9

Comparing Fractions using Benchmark Values

Estimate values using benchmarks. For example: What is 10/22 of 5/18 of 2000? If you recognize that 10/22 is nearly 1/2 and 5/18 is approx. 1/4 then it is easier to determine. Try to make rounding errors cancel by rounding some numbers up and other numbers down

Signs of Square Roots

Even roots only have a positive value. A root can only have a negative value if (1) it is an odd root and (2) the base of the root is negative

Fewer Factors More Multiples

Factors divide into an integer and are therefore less than or equal to that integer. Positive multiples on the other hand multiply out from an integer and are therefore greater than or equal to that integer

Complex Absolute Value

For a problem with two different variables, generally without constants, are more easily solved using a conceptual approach rather than algebraic. Try picking and testing numbers, specifically positives, negatives and zero

Strategy for solving data sufficiency-test numbers

For a value question, try to find multiple answers. For a yes/no question try to find a maybe. Be sure to try a positive, negative, integera and fractional number unless explicitly told otherwise.

Sum of consecutive integers and divisibility

For any set of consecutive integers with an ODD number of items, the sum of all the integers is always a multiple of the number of items. For example: 4+5+6+7+8= 30 which is a multiple of 5 For any set of consecutive integers with an EVEN number of items, the sum of all the integers is never a multiple of the number of items.

Fraction Operations:Funky Results

For proper fractions: Adding Fractions --> Increases the value Subtracting Fractions --> Decreases the value Multiplying Fractions --> Decreases the value Dividing Fractions--> increases the value

When to use Fractions vs. decimals

Fractions are good for cancelling factors in multiplications or expressing proportions that do not have clean decimal equivalents. Decimals are good for estimating results or for comparing sizes. Prefer fractions for doing multiplication/division but prefer decimals and percents for doing addition or subtraction, estimating numbers or comparing numbers

Estimating Roots of Imperfect Squares

If there is no coefficient in front you may estimate by figuring the two closest perfect squares on either side of it. If you want to estimate a square root with a coefficient, simply estimate the square root and then multiply by the coefficient. Or combine the coefficient with the root.

Multiply Decimals

Ignore the decimal point until the end. Just multiply the numers as if they were whole numbers. Then count the total number of digits to the right of the decimal point in the factors. In the factors, count all the digits to the right of the decimal point--then put that many digits to the right of the decimal point in the product If multiplying a very large number and a very small number, move the decimals in the opposite direction the same number of places.

Simplifying Square Roots

You may only seperate or combine the product or quotient of two roots. You cannot seperate or combine the sum or difference of two roots

Area of Trapezoid

[(Base 1 + Base 2) x Height] / 2 Average of two bases multiplied by the height

The Last digit Shortcut

When asked to find the units digit, just look at the units digit of the product. For example: What is the units digit of (7^2)(9^2)(3^3)? Step 1: 7 x 7 = 49 - Drop all except units digit - 9 Step 2: 9 x 9 = 81 - Drop all except units digit - 1 Step 3: 3 x 3 x 3 = 27 - Drop all except units digit - 7 Step 4: 9 x 1 x 7 = 63 The units digit of the final product is 3

Subtracting Exponents

When dividing two terms with the same base, combine exponents by subtracting

Adding Exponents

When multiplying two terms with the same base, combine exponents by adding

Nested Exponents

When raising a power to a power, combne exponents by multiplying

Percent shortcuts

When trying to find a more complicated percentage, break it into easy to find chunks. For example: 23% of 400: 10% of 400 is 40 therefore 20% is 2 x 40 = 80. 1% of 400 is 4 and 3% is 3 x 4 or 12. Putting it together, we get 80+12=92

Range of possible remainders

When you divide an integer by 7, the remainder could be any number between 0 and 6 inclusive. Notice that you cannot have a negative remainder or remainders larger than 7. There are exactly 7 possible remainders.

An Exponent of 1

When you see a base without an exponent, write in an exponent of 1

Fractional Base Exponents

While most positive numbers increase when raised to a higher exponent, numbers between 0 and 1 decrease

Fractional Exponents

Within the exponent fraction, the numerator tells us what power to raise the base to, and the denominator tells us which root to take

Arithmetic with remainders

You can add and subtract remainders as long as you correct excess or negative remainders

Multiplication with remainders

You can multiply remainders as long as you correct excess remainders at the end. For example, if x has a remainder of 4 upon division by 7 and z has a remainder of 5 upon division by 7, then 4 x 5 gives 20. Two additional 7's can be taken out of this remainder, so xz will have remainder 6 upon division by 7.

Complex Abs Value with more than one expression

an absolute value with more than one expression but only one variable and one or more constants is usually easier solved with an algebraic approach

Repeating Decimals

divide any number by 9 and it becomes a repeating decimal. For example: 4/9 = 0.44444444 forever. 3/11 = 27/99 = 0.272727272727 forever

Property of GCF-consecutive multiples

have a GCF of n. For example, 8 and 12 are consecutive multiples of 4. Thus 4 is a common factor of both numbers. But 8 and 12 are exactly 4 units apart. Thus 4 is the greatest possible common factor of 8 and 12

Divisible by 5 if

integer ends in 5 or 0

Divisible by 4 if

integer is divisible by 2 twice or the last two digits are divisible by 4

Divisible by 6 if

integer is divisible by both 2 and 3

Add/Subtract a multiple with a nonmultiple

result is a nonmultiple Multiple of 3 + Nonmultiple of 3 = Nonmultiple of 3

Add/Subtract two nonmultiples

results can be multiple or nonmultiple

Divisible by 9 if

sum of the integer's digits are divisble by 9

Prime Factorization Factors

the prime factorization of a perfect square contains only even powers of primes. Vice versa.

Products of Consecutive Integers and Divisibility

the product of n consecutive integers is divisible by n!

Multiplying & Dividing Signed Number

the result will be positive if you have an EVEN number of negative numbers in the collection. The result will be negative if you have an ODD number of negative numbers.