Chapter 3: Properties of Numbers

First 9 Perfect Squares

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

First 9 Perfect Cubes

0, 1, 6, 27, 64, 125, 216, 343, 512

Finding the Number of Factors in a Particular Number

1) Find the prime factorization of the number 2) Add 1 to the value of each exponent, then multiply these results and the product will be the total number of factors for that number

Finding the LCM

1. Find the prime factorization of each integer 2. Of any repeated prime factors among the set, take only those with the largest exponent (only take each number once) 3. Of what is left, take all of the non-repeated prime factors 4. Multiply together what you found in steps 2-3

Finding the GCF

1. Find the prime factorization of each number 2. Identify repeated prime factors 3. Of any repeated prime factors among the numbers, take only those with the smallest exponent (if none are repeated, the GCF=1) 4. Multiply together the numbers from step 3

Properties of One (6 properties)

1. One is a factor of all numbers, and all numbers are multiples of one 2. One raised to any power = 1 3. Multiplying or dividing by 1 does not change the number 4. One is an odd number 5. One is the only number with exactly 1 factor 6. One is not a prime number (the first prime number is 2)

Using Trailing Zeros to Determine the Number of Digits in an Integer

1. Prime Factor the number 2. Count the number of (5x2) pairs -- each contributes a trailing zero 3. Collect the number of unpaired 5s and 2s along with other nonzero prime factors and multiply them together and count the number of digits 4. Sum the number of digits from steps 2-3

Properties of Zero (13 properties)

1. Zero multiplied by any number = 0 2. Zero divided by any number other than zero = 0 3. Any number divided by zero = undefined 4. The square root of zero = 0 5. Zero raised to any positive power = 0 6. Zero is the only number that is neither positive nor negative 7. Zero is the only number that equals its opposite 8. Zero is a multiple of all numbers 9. Zero is the only number that equals all of its multiples 10. Zero is not a factor of any number, except 0 11. Any number (except 0) raised to the zero power = 1 12. Zero is an even number 13. Zero can be added/subtracted to any number without changing the value

Prime Numbers under 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

Integer

A number that can be written without a decimal or fractional component Whole numbers: nonnegative integers (including 0)

Absolute Value

A number's distance from zero on the number line

Greatest Common Factor (GCF)

A common factor of a set of positive integers is any number that can divide into all of the numbers in the set

#10 Divisibility Rule (0-12)

A number is divisible by 10 if the ones digit is a zero

#11 Divisibility Rule (0-12)

A number is divisible by 11 if the sum of the odd-numbered place digits minus the sum of even-numbered place digits is 11

#12 Divisibility Rule (0-12)

A number is divisible by 12 if the number is divisible by both 3 and 4

#2 Divisibility Rule (0-12)

A number is divisible by 2 if the ones digit is 0, 2, 4, 6, 8 (even)

#3 Divisibility Rule (0-12)

A number is divisible by 3 if the sum of all of the digits is divisible by 3

#6 Divisibility Rule (0-12)

A number is divisible by 6 if the number is an even number whose digits sum to a multiple of 3

#8 Divisibility Rule (0-12)

A number is divisible by 8 if the number is even, divide the last three digits by 8 - if there is no remainder it is divisible *(remember that 000 is divisible by 8)

#9 Divisibility Rule (0-12)

A number is divisible by 9 if the sum of all digits is 9

Prime Factorization of Perfect Cubes

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

Divisibility

A positive integer x is divisible by some positive integer y if y divides into x without leaving a remainder * Easiest way to understand divisibility is to use prime factorization

Repeated Prime Factors

A prime factor is repeated when it is shared by 2 or more numbers in the set, it does not have to be shared by all to be repeated Be careful of this when finding the LCM

The Range of Possible Remainders

A remainder must be a non-negative integer that is less than the divisor divisor = n remainder = any integer 0 -> (n-1)

Excess Remainder

A remainder that is greater than its divisor Can correct this by removing as many multiples of the original divisor as we can from the excess remainder

Even & Odd Numbers

All integers are either even or odd Even: divisible by 2, all have even units digits (0,2,4,6,8) Odd: not divisible by 2, all have odd units digits (1,3,5,7,9) Even integers: 2n Odd integers: 2n-1, 2n+1

#1 Divisibility Rule (0-12)

All numbers are divisible by 1

The Factor Chart

All positive integers have a finite number of factors, so you can use a factor chart to find all of the factors of a number

#1 Units Digit Pattern

All powers of 1 end in 1

#0 Units Digit Pattern

All powers of zero end in 0

#2 Units Digit Pattern

All units digits of positive powers of 2 will follow the pattern 2-4-8-6

#3 Units Digit Pattern

All units digits of positive powers of 3 will follow the pattern 3-6-7-1

#4 Units Digit Pattern

All units digits of positive powers of 4 will follow the pattern 4-6

#5 Units Digit Pattern

All units digits of positive powers of 5 will end in 5

#6 Units Digit Pattern

All units digits of positive powers of 6 will end in 6

#7 Units Digit Pattern

All units digits of positive powers of 7 will follow the pattern 7-9-3-1

#8 Units Digit Pattern

All units digits of positive powers of 8 will follow the pattern 8-4-2-6

#9 Units Digit Pattern

All units digits of positive powers of 9 will follow the pattern 9-1

Prime Factorization

Any composite number can be broken down and expressed as the product of its prime factors Use a prime factorization tree, then collect all of the prime factors and multiply them

Prime Number

Any integer greater than 1 that has no factors other than 1 and itself - Composite numbers are the opposite - 2 is the only even prime number

Determining the Number of Trailing Zeros

Any multiplication fo 5 and 2 will create a units digit of 0, regardless of what else is being multiplied Trailing zeros are created by powers of 10 -- each (5x2) pair creates a trailing zero *Any factorial greater than 5! will always have 0 as its units digit

Prime Factors v. Unique Prime Factors

If a number has two unique prime factors, that means that number has two prime factors that are not equal to each other If we say a number has two prime factors, we mean that number has a total of two prime factors but they are not necessarily unique

Prime Factorization of a Perfect Square

If some number x (that is not 0 or 1) is a perfect square, the prime factorization of x must contain only even exponents

#5 Divisibility Rule (0-12)

If the last digits are 0 or 5 the number is divisible by 5

Determining the Number of Leading Zeros in the Decimal Equivalent of a Fraction

Express the fraction in the form of 1/x, where x is an integer If 1 < x <= 10, no leading zeros If 10 < x <= 100, 1 leading zero If x is an integer with K digits, then 1/x will have (k-1) leading zeros, unless X is a perfect power of 10, then it will have (k-2)

#4 Divisibility Rule (0-12)

If the last two digits of a number are divisible by 4, then the number is divisible by 4

Factors

For any positive integer x and y, y is a factor of x if and only if x/y is an integer 1 <= y <= x

Perfect Square

If the square root of some integer x is an integer, then x is a perfect square The square root of a perfect square is always a whole number A number that ends in 2,3,7,8 can never be a perfect square

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

If we know that y divides evenly into x, then: LCM of x and y = y GCF of x and y = y

Known LCM and GCF

If we know the LCM and GCF of x and y, the product of x and y can be found by multiplying the LCM and GCF together (x)(y) = (LCM)(GCF)

Prime Factors from Known LCM

If we know the LCM of a set of positive integers, we can find all of the unique prime factors of the numbers in the set In determining the LCM, we must multiply the repeated prime and non-repeated prime factors (which is the entire list of unique prime factors)

Factor of Factors

If x is divisible by y, then x is also divisible by all factors of y

If z is divisible by both x and y, then...

If z is divisible by x and y, z must also be divisible by the LCM of x and y (but not necessarily further multiples)

#0 Divisibility Rule (0-12)

No numbers are divisible by 0

#7 Divisibility Rule (0-12)

No rules, just do the division

Even Division

Occurs when the numerator of a positive fraction is a multiple of the denominator

Signed Numbers

Positive or negative number

Using the LCM to Solve Repeating Pattern Questions

The LCM can be used to determine when two processes that occur at differing rates or times will coincide

Remainder

The amount left over when one number is divided by another

Determining Terminating Decimals through Prime Factorization

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

The number of unique prime factors in a number does not change when that number is raised to a positive integer exponent

The number of unique prime factors in a number remains constant even when the number is raised to a positive integer power

Evenly Spaced Sets

The numbers in the set increase by the same amount and therefore share a common difference

Multiplication Rules for Even and Odd Numbers

The product of an even number and any integer is always even [(even)(even) = even , (even)(odd) = even] The product of an odd number and any odd number is always odd [(odd)(odd) = odd]

Division Properties of Factorials

The product of any set of consecutive integers is always divisible by any of the integers in the set - and also divisible by any of the factor combinations of the numbers * The product of any set of n consecutive positive integers is divisible by n!

Multiples

The product of that number with any integer - An infinite number of multiples exist for every number - To find a multiple of any quantity x, multiply x by any integer

Division Rules for Even and Odd Numbers

The rules of division only apply if one integer divides evenly into the other Even/Even = Even or Odd Even/Odd = Even Odd/Odd = Odd

Least Common Multiple (LCM)

The smallest positive integer into which all of the numbers in the set will divide (the smallest positive multiple)

Leading Zeros in Decimals

The zeros that occur to the right of the decimal point but before the first nonzero number

Converting a Remainder from Fraction to Decimal

To convert a fractional remainder into a decimal remainder, we only need to convert the fraction to a decimal

Converting a Decimal Remainder to an Integer

To determine what the integer remainder is, we can multiply the decimal component of the result of the division by the divisor (but need to know the denominator)

Consecutive Integers and Prime Factors

Two consecutive integers will never share the same prime factors (the GCF of any consecutive integers is 1)

Patterns in Units Digits (0-9)

When a positive integer is raised to consecutive powers that are positive integers a pattern will arise in the units digit of the product

No Shared Prime Factors

When a set of positive integers shares no prime factors, the LCM of that set will be the product of the numbers in that set

Remainders after Division by 10^n

When a whole number is divided by 10, the remainder will be the units digit of the dividend (numerator) Divided by 100, the remainder will be the last two digits, etc.

Addition when signs are different

When adding numbers with opposite signs, the result is always between the two addends Subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the greater absolute value

Remainders after Division by 5

When integers with the same units digit are divided by 5, the remainder will always be the same

Multiplication and division of numbers with different signs

When multiplying or dividing two numbers with different signs, the product or quotient with always be negative

Multiplication and division of numbers with the same sign

When multiplying or dividing two numbers with the same sign, the product or quotient with always be positive

Converting a Remainder from Decimal to Fraction

When the division of two integers, x and y, results in a decimal we can't be sure what the actual fractional remainder is without knowing the values of x and y But we can determine what the most reduced fraction could be, and the actual remainder will be a multiple of this

Addition and Subtraction Rules for Even and Odd Numbers

When we add or subtract two numbers: - The sum/difference will be even only if both numbers are even or both numbers are odd - Otherwise, the sum/difference will be odd

Addition when both numbers are negative

When we add two negative numbers, the result is a smaller negative number

Addition when both numbers are positive

When we add two positive numbers, the result is a larger positive number

Divisibility with Exponents

When we divide like bases, we keep the base and subtract the exponents

Subtraction of Signed Numbers

When we subtract a number, we add its opposite (and use the addition rules)

Perfect Cubes

When we take the cube root of a non-negative integer x, if the result is an integer, then x is a perfect cube

Formula for Division

x/y = Q + (r/y) x = Qy + r Q = (x-r)/y r = x - Qy Q = quotient R = remainder