GMAT Number Properties

Divisibility Rules 1

- 2, integer is EVEN - 3, SUM of integers digits is divisible by 3 - 4, divisible by 2 twice OR if last 2 digits are divisible by 4 - 5, integer ends in 0 or 5 - 6, divisible by both 2 AND 3 - 8, divisible by 2 three times OR last 3 digits divisible by 8 - 9, sum of digits divisible by 9 - 10, number ends in 0

Relationships in Even Sets, Consecutive Multiples and Integers

- 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 these three parameters are known: 1. The smallest (first) or largest (last) number in the set 2. The increment (always 1 for consecutive integers 3. The number of items in the set

Factor Pairs

- pairs of factors that yield an integer when multiplied together - how to find a factor pair of X: 1. Make a table with 2 columns labeled "small" and "large" 2. Start with 1 in the small column and X in the large 3. Test the next possible factor of X 4. Repeat until the numbers in the small and large columns converge

Primes to 100

1 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

Properties of GCF & LCM

1. (GCF of M&N) * (LCM of M&N) = M*N 2. GCF of M&N cannot be greater than M-N 3. Consecutive multiples of N have a GCF of N. - GCF of two consecutive integers is 1

Exponent Rules

1. Adding: when multiplying two numbers with the same base, combine exponents by adding 2. Subtracting: when dividing two numbers with the same base, combine exponents by subtracting them 3. Multiplying: when raising a power to a power, combine exponents by multiplying 4. Dividing: a negative exponent means putting the term with the exponent as the bottom of a fraction and making the exponent positive - when there is a negative exponent, think reciprocal 5. Exponent of 1: any number that doesn't have an exponent implicitly has an exponent of 1 6. Exponent of 0: any non-zero base raised to the power of 0 is equal to 1 7. Fractional exponents: the numerator tells us the POWER to raise the base to, and the denominator tells us which ROOT to take

Perfect Squares

1. All perfect squares have an odd number of total factors - any integer that has an odd number of factors MUST be a perfect square. 2. The prime factorization of a perfect square contains only even powers of primes. - any number whose prime factorization contains only even powers of primes must be a perfect square. 3. Same rules apply to perfect cubes, quads,etc., and factorization needs to have multiples of that "perfect N".

Properties of Evenly Spaced Sets

1. Arithmetic mean = median. - if there are an even number of elements, the median is the average of the two middle elements 2. Mean & median = average of the first and last element. 3. Sum of elements = mean * number of items in the set.

Creating Prime Columns

1. Calculate the prime factors of each integer. 2. Create a column for each prime factor found within any of the integers. 3. Create a row for each integer. 4. In each cell, place the prime factor raised to a power. This power counts how many copies of the column's prime factor appear in the prime box of that row's integer.

Simplifying Roots 1

1. Combining/separating in multiplication and division, NEVER in addition or subtraction. 2. How to simplify? - multiplying or dividing: split into factors or simplify into a single root of the product. 3. Imperfect squares: - simplify into primes then take out smaller roots

Counting Integers

1. Consecutive integers: "add one before you're done" (Last - First) + 1 2. consecutive multiples: (Last - First)/Increment + 1

When to use Prime Factorization

1. Determine if x is divisible by y 2. Determine the greatest common factor of two numbers 3. Reducing fractions 4. Finding the least common multiple of a set of numbers 5. simplifying square roots 6. Determine the exponent of one side of an equation with integer constraints

Sums of Consecutive Integers & Divisibility

1. For any set of consecutive integers with an ODD number of items, the sum of all integers is ALWAYS a multiple of the number of items. - ex: 1+2+3+4+5=15, multiple of 5 2. for any set of consecutive integers with an EVEN number of items, the sum of all items is NEVER a multiple of the number of items. - ex: 1+2+3+4=10, not a multiple of 4 3. Use prime boxes to keep track of factors of consecutive integers.

How to Find GCF & LCM with Prime Columns

1. GCF = the lowest count of each prime factor found across ALL integers (counts the shared primes). 2. LCM = the highest count of each prime factor found across ALL integers (counts the primes less the shared primes). 3. For any integer A & B, the GCFxLCM = AxB 4. Can use the same technique on prime boxes.

Estimate Roots of Imperfect Squares

1. If a simple square root (no coefficient), figure out two closest perfect squares on either side and estimate between those roots. 2. If a coefficient, estimate as step one and multiply by coefficient OR combine coefficient with the root and then do step 1

Divisibility Rules 2

1. If you add a multiple of N to a non-multiple of N, the result is a non-multiple of N (same for subtraction). 2. If you add two non-multiples of N, the result could either be a multiple or non-multiple of N.

Factorials

1. N! Is the product of all positive integers smaller than or equal to N. 2. Works for multiples of two numbers if they share factors. 3. Any smaller factorial divides into any larger factorial - the smaller factor cancels completely.

Simplifying Exponential Expressions 1

1. Only if expressions are linked by multiplication or division. CANNOT simplify expressions linked by addition or subtraction. 2. Only if linked by multiplication or division if they have a base or exponent in common. 3. Use exponent rules (book 1, pg. 68). 4. Factor whenever bases are the same. 5. Factor when the exponent is the same and the terms have something in common. 6. Study chart in book 1, pg. 70.

Data Sufficiency Strategy

1. Rephrase the question whenever possible. 2. Factor algebraic expressions when possible. 3. Focus on how the piece of information relates to the question. 4. The answer must yield a specific value or a YES/NO answer. 5. Test numbers that give a MAYBE or multiple values, then you know a statement is not sufficient - consider positive, negative, integers, fractions 6. The two statements given will NEVER contradict each other.

Primes 2

1. There is an infinite number of primes. 2. There is no simple pattern to prime numbers. 3. Positive integers with only two factors must be primes. 4. Positive integers with two or more factors is NEVER prime. 5. 1 is not a prime number. 6. 2 is the only even prime.

Roots

1. When GMAT does a square root or an even root, they are only looking for the positive root. 2. For odd roots, the answer will be the same sign as the base. 3. Study pages 81-82 for comment roots and exponents and root rules.

Sum of two Primes

ALL primes are odd except for 2 - sum of any two primes will always be even unless one of the primes is 2 - if you see a sum of two primes that is odd, one number must be 2

Odds & Evens

Addition/Subtraction - 2 odds or evens = EVEN - odd with an even = ODD Multiplication - if any integer is even = EVEN - none integers is even = ODD - the more evens there are, divisible by increasing powers of 2 (2 integers div. by 4, 3 integers div. by 8)

Divisibility & Prime Strategy

All of the following say the same thing: - X is divisible by Y = Y is a divisor of X - X is a multiple of Y = Y divides X - X/Y is an integer = X/Y yields a remainder of 0 - X=3(n), n being an integer = Y "goes into" X evenly

Primes 1

Any positive integer larger than 1 with exactly two factors: 1 and itself. - 1 is not considered a prime - the first prime is 2, only even prime - first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

"Fewer Factors, More Multiples"

Factors divide into an integer "X" and are therefore less than or equal to X. Positive multiples multiply out from X and are therefore greater than or equal to X. - all integers have a limited number of factors - all integers have an infinite number of multiples

Simplifying Exponential Expressions 2

Four step process: 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.

Bases of Exponents 2

Fractional: if positive proper fraction, as the exponent increases, the value of the expression decreases - increasing powers cause fractions to decrease - decimals between 0 and 1 are the same Compound base: - when a product, multiply base first then raise to power OR distribute the exponent first then multiply - when a sum, MUST add the numbers in parentheses first

Greatest Common Factor (GCF) & Least Common Multiple (LCM)

GCF: the largest divisor of 2+ integers LCM: the smallest multiple of 2+ integers - use a venn diagram to find GCF and LCM 1. Factor both numbers into primes 2. Place common factors in the shared area (incl. copies) 3. Place non-common factors in the sides 4. GCF = product of primes in the shared area 5. LCM = product of all primes in the diagram - if no primes are in common, the GCF is 1 and the LCM is the product of the two numbers

Use Conjugates to Rationalize Denominators

Get rid of square roots in the denominator of a simple expression by multiplying the numerator and denominator by that square root. For complex expressions, use the conjugate of the denominator. - change the sign of the square root term

Prime Box

Holds all prime factors of N. - must repeat copies of prime factors if N has multiple copies of that prime factor - can tell if X is a factor of Y by multiplying factors in the prime box - to find divisors of N, N must also be divisible by all possible products of the primes in the box

Factor Foundation Rule

If A is a factor of B, and B is a factor of C, then A is a factor of C. - an integer is divisible by all its factors and all factors of its factors - doesn't include 1

Multiplying & Dividing Signed Numbers

If Signs are the Same, the answer is poSitive but if Not, the answer is Negative. - when multiplying or dividing a group of non-zero numbers, the result will be positive if you have an EVEN number of negative numbers. - the result will be negative if you have an ODD number of negative numbers

Divisibility & Addition/Subtraction

If you add or subtract Multiples of N, the result is a multiple of N. - N is the divisor of x and y, then N is a divisor of x + y - ex: 35 + 21 = 56

Arithmetic with Remainders

If you have the same divisor throughout: 1. You can add/subtract remainders directly as long as you correct excess or negative remainders. - excess = remainders greater than or equal to the divisor - to correct, add or subtract the divisor 2. You can multiply remainders as long as you correct excess remainders at the end.

Multiple

Integer formed by multiplying that integer by any integer. - negative multiples are possible - zero is a multiple of every number - every integer is a multiple of itself

Divisibility of Odds & Evens

NO GURANTEES! - can result in odds, evens or non-integers - odd divided by any number will never be even - odd divided by even will never equal an integer

Remainders 1

On simple problems, pick numbers. - add the desired remainder to a multiple of the divisor - ex: need a number that leaves a remainder of 4 after dividing by 7; (7*2) + 4 = 18

Factor

Positive integer that divides evenly into an integer. - every integer is a factor of itself - 1 is a factor of every integer

Evenly Spaced Sets

Sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next - ex: 4,7,10,13,16

Bases of Exponents 1

Sign: when negative, simply multiply as required; beware when there is an EVEN exponent. - hides the sign of the base - any base raised to an even power is a positive answer - odd exponents always keep the sign of the base - base of 0, 1, -1 ~ base of 0=0 ~ base 1=1 ~ base -1=1 or -1

Consecutive Integers

Special cases of consecutive multiples: all the values in the set increase by 1, all integers are multiples of one - ex: 12,13,14,15,16

Consecutive Multiples

Special cases of evenly spaced sets: all values in the set are multiples of the increment - ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4 - these sets must be composed of integers

Absolute Value

Tells how far a number is from 0 on the number line and is always positive. - if two numbers are opposite each other, they have the same absolute value

Absolute Value of a Difference

The absolute value of the difference between x & y is the distance between x & y on the number line. Absolute value equations: 1. The equation/inequality contains 2 different variables in absolute value expressions - use a more conceptual approach. 2. The equation/inequality has more than 1 absolute value expression but only 1 or more constants - use an algebraic approach.

Products of Consecutive Integers

The product of K consecutive integers is always divisible by K factorial (K!) - ex: 3! = 3x2x1 = 6, always divisible by 3&2

Prime Factorization

To find primes, create a prime factor tree. - factors of N can be found by building all possible products of the prime factors

Counting Total Factors

To find the total number of factors in a large number: 1. Break a number into it's prime factorization. 2. Add a one to each prime's exponent. 3. Multiply all the results of #2 together to get the number of different factors.

Simplifying Roots 2

Using prime factorization: 1. Factor the number under the radical sign into primes. 2. Pull out any pair of matching primes from under the radical sign, and place one of those primes outside the root. 3. Consolidate the expression.

Factoring & Distributing Exponents

When you encounter any exponential expressions in which two or more terms in include something common in the base, consider factoring. When an expression is given in factored form, consider distributing it. - study book 1, pg. 164

Disguised Positive & Negative Questions

Whenever you see inequalities with zero on either side of the inequality, you should consider testing positive/negative cases to help solve the problem.

Integers

Whole numbers that are either positive or negative, including zero. - adding, subtracting and multiplying result in integers. - result of division is called a quotient. - "divisible" = no remainders - "divisor" or "factor" is a number that can evenly be divided into another

Remainders 2

x = Q*N + R - x= dividend - Q=quotient - N=divisor - R=remainder *all must be integers The remainder of any number must be non-negative and smaller than the divisor. When you divide by a positive integer N, there are N possible remainders.