Mission 6: Properties of Numbers

Divisibility rules

0 = No number is divisible by 0 1 = All numbers are divisible by 1 2 = A number is divisible by 2 if the units digit is even 3 = A number is divisible by 3 if the sum of its digit is divisible by 3 4 = A number is divisible by 4 is the last two digits of a number is divisible by 4 5 = A number is divisible by 5 if the units digit is 0 or 5 6 = A number is divisible by 6 if the number is even whose sum of its digits is divisible by 3 7 = Just do the division 8 = A number is divisible by 8 if the number is even whose last 3 digits are divisible by 8 9 = A number is divisible by 9 if the sum of its digits is divisible by 9 10 = A number is divisible by 10 if it ends in 0 11 = A number is divisible by 11 if the sum of the odd place digits - sum of the even place digits is divisible by 11

Patterns in units digits (positive integer exponent patterns)

0: All powers of 0 end in 0 1: All powers of 1 end in 1 2: The units digits follow the pattern: 2-4-8-6 3: The units digits follow the pattern: 3-9-7-1 4: The units digits follow the pattern 4-6 (odd powers = 4, even powers = 6) 5: All powers of 5 end in 5 6: All powers of 6 end in 6 7: The units digits follow the pattern: 7-9-3-1 8: The units digits follow the pattern 8-4-2-6 9: The units digits follow the pattern 9-1 (odd powers = 9, even powers = 1)

Finding the number of factors of a particular number

1. Find the prime factorization 2. Add 1 to the value of each exponent and multiply the results. The product will be the total number of factors of that number

Process for finding the LCM

1. Prime factorize each integer 2. Of any repeated prime factors, take only those with the largest exponent. If left with two of the same power, just take that number once 3. Of what is left, take all non-repeated prime factors 4. Multiply together what was found in steps 2 and 3, the result is the LCM

Process for finding the GCF

1. Prime factorize each number 2. Identify repeated prime factors 3. Of any repeated prime factors among the numbers, take only those with the smallest exponent 4. Multiply the numbers found in step 3, this result is the GCF

Using trailing zeros to determine the number of digits in an integer

1. Prime factorize the number 2. Count the number of (5x2) pairs, each contributes to one trailing zeros 3. Collect the number of unpairs 5s or 2s, along with other nonzero prime factors and multiply them together. Count the number of digits in this product 4. Sum the number of digits from steps 2 and 3

Prime numbers less than 100

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

The product of any set of n consecutive positive integers is divisible by n!

5x6 is divisible by 2! 7x8x9 is divisible by 3! 3x4x5x6 is divisible by 4!

Multiples

A multiple of a number is the product of that number with any integer

The prime factorization of a perfect cube will contain only exponents that are multiples of 3

A perfect cube, other than 0 and 1, is a number such that all of its prime factors have exponents that are divisible by 3

The prime factorization of a perfect square will contain only even exponents

A perfect square, other than 0 or 1, is a number such that all of its prime factors have even exponents

Any factorial >= 5! will always have zero as its units digit

Any factorial 5! and above will have at least one (5x2) pair

Addition and subtraction rules for even and odd numbers

Even + even = even Odd + odd = even Even + odd = odd Odd + even = odd

Division rules for even and odd numbers

Even / odd = even Odd / odd = odd Even / even = even or odd

Multiplication rules for even and odd numbers

Even x even = even Even x odd = even Odd x odd = odd

Factors

For any positive integers x and y, y is a factor of x if and only if x/y is an integer

If a set of positive integers has no prime factors in common, the GCF is 1

For any set of positive integers, the LCM will always be greater than or equal to the largest number in the set, and the GCF will always be less than or equal to the smallest number in the set

The product of two positive integers may or may not be the LCM of those

If two positive integers share no prime factors, the LCM is the product of the two integers Otherwise, the LCM is some number less than their product

The LCM and GCF when one number divides evenly into the other

If we know the positive integer y divides evenly into positive integer x, the LCM of x and y is x and the GCF is y Ex: LCM of 25 and 100 = 100, GCF = 25

Factors of factors

If x and y are positive integers and x/y is an integer, then x/any factor of y is also an integer

Patterns in units digits greater than 9

Integers greater than 9 have the same units-digit pattern as the powers of its units digit

Prime numbers

Prime numbers only have two factors: 1 and themselves 2 is the only even and is the smallest prime number

Adding and subtracting remainders

Remainders can be added, but we must correct for excess remainders Remainders can be subtracted, but we must correct for negative remainders When we add or subtract remainders, we need to keep them in order

Two consecutive integers will never share the same prime factors

The GCF of two consecutive integers is 1

if z is divisible by both x and y, z must also be divisible by

The LCM of x and y

Perfect cubes

The cube root of a perfect cube will always be an integer. In taking the cube root of a nonnegative integer x, if the result is an integer, then x is a perfect cube The first nine perfect cubes are: 0, 1, 8, 27, 64, 125, 216, 343, 512

Terminating decimals

The decimal equivalent of a fraction will terminate if and only if the denominator of the reduced fraction has a prime factorization that contains only 2s or 5s, or both If the prime factorization of the reduced fraction's denominator contains anything other than 2s or 5s, the decimal equivalent will not terminate

GCF

The largest number that will divide into all of the numbers in the set

Determining the number of trailing zeros

The number of trailing zeros of a number is the number of (5x2) pairs in the prime factorization of that number

Even division

The numerator of a positive fraction is a multiple of the denominator

Division properties of factorials

The product of any set of consecutive integers is divisible by any of the integers in the set and by any of the factor combinations of those numbers

LCM

The smallest positive multiple of all integers in the set

Perfect squares

The square root of a perfect square will always be an integer. In taking the square root of a nonnegative integer x, if the result is an integer, then x is a perfect square Perfect squares cannot end in 2, 3, 7, or 8 The first nine perfect squares are: 0, 1, 4, 9, 16, 25, 36, 49, 64

Consecutive integers

There are three ways evenly spaced sets commonly appear on the GMAT: 1. A set of consecutive integers (including sets of even and odd integers) 2. A set of consecutive multiples of a given number 3. A set of consecutive numbers with a given remainder when divided by some integer

Converting a decimal remainder into an integer

To determine the integer remainder, we can multiply the decimal component of the result of the division by the divisor to get the remainder Ex: 0.8x5 = 4 -> the remainder is 4 Need to know the denominator to use this

Shortcut for determining the number of primes in a factorial when the base of the divisor is not a prime number

To determine the largest number of a prime number x that divides into y! 1. Break x into prime factors 2. Using the largest prime factor of x, apply the factorial divisibility shortcut to determine the quantity of that prime factor. The quantity determined represents the largest number of x that divides into y!

Shortcut for determining the number of primes in a factorial

To determine the largest number of a prime number x that divides into y! 1. Divide 1 by x^!, x^2, x^3, etc. keep track of the quotients while ignoring remainders. Stop dividing when y/x^n 2. Add the quotients from the previous divisions, that sum is the number of prime number x

Shortcut for determining the number of primes in a factorial when the base of the divisor is a power of a prime number

To determine the largest number of a prime number x that divides into y! 1. Express x = p^k (p is prime, k is an integer greater than 1) 2. Apply the factorial divisibility shortcut to determine the quantity of p in y! Then create and simplify an inequality to determine the largest number of x that divides into y!

Even and odd exponents vs. positive and negative answers

When a nonzero base is raised to an even exponent, the result will always be positive When a nonzero base is raised to an odd exponent, the result will depend on the base (if the base is positive the result will be positive, if the base is negative the result will be negative)

Remainders after division by 10^n

When a whole number is divided by 10, the remainder will be the units digit of the dividend, etc. Ex: 153/10 = 15.3, remainder = 3

Remainders after division by 5

When integers with the same units digit are divided by 5, the remainder is constant

Determining the number of leading zeros in the decimal equivalent of a fraction in the form of 1/x, when x is an integer

When the denominator x is not a perfect power of 10 -> 1/x will have K-1 leading zeros When the denominator x is a perfect power of 10 -> 1/x will have K-2 leading zeros K = number of digits

Converting a remainder from decimal form to fractional form

When the division of two integers results in a decimal, we can't be sure of the actual remainder The actual remainder, however, must be a multiple of the most reduced fractional remainder

Multiplying remainders

You can multiply remainders as long as you correct excess remainders at the end. If x has a remainder of 4 upon division by 7 and 2 has a remainder of 5, 4 by 5 is 20. Two 7's can be taken out so that x by z will have a remainder of 6

A formula for division

x/y = Quotient + remainder/y