FUNDAMENTAL ARITHMETIC Prime Numbers Prime numbers are any whole numbers greater than 1 that can only be divided by 1 and itself. Below is the list of all prime numbers between 1 and 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 Prime Factorization Prime factorization is the process of changing non-prime (composite) numbers into a product of prime numbers. Below is the prime factorization of all numbers between 1 and 20. 1 is unique 6 = 2 ∙ 3 11 is prime 16 = 2 ∙ 2 ∙ 2 ∙ 2 2 is prime 7 is prime 12 = 2 ∙ 2 ∙ 3 17 is prime 3 is prime 8 = 2 ∙ 2 ∙ 2 13 is prime 18 = 2 ∙ 3 ∙ 3 4 = 2 ∙ 2 9 = 3 ∙ 3 14 = 2 ∙ 7 19 is prime 5 is prime 10 = 2 ∙ 5 15 = 3 ∙ 5 20 = 2 ∙ 2 ∙ 5 Divisibility By using these divisibility rules, you can easily test if one number can be evenly divided by another. A Number is Divisible By: If: Examples 2 The last digit is 5836 divisible by two 583 ퟔ ퟔ ÷ 2 = 3, Yes 3 The sum of the digits is 1725 divisible by three 1 + 7 + 2 + 5 = ퟏퟓ ퟏퟓ ÷ 3 = 5, Yes 5 The last digit ends 885 in a five or zero 88 ퟓ The last digit ends in a five, Yes 10 The last digit ends 7390 in a zero 739 ퟎ The last digit ends in a zero, Yes Exponents An exponent of a number says how many times to multiply the base number to itself. 3 ←푒푥푝표푛푒푛푡 5 6 5 ↑ 2 7 푏푎푠푒 5⏟ × 5 × 5 2⏟ × 2 × 2 × 2 × 2 × 2 7⏟ × 7 × 7 × 7 × 7 푡ℎ푟푒푒 5푠 푚푢푙푡푝푙푒푑 푡표푔푒푡ℎ푒푟 푠푥 2푠 푚푢푙푡푝푙푒푑 푡표푔푒푡ℎ푒푟 푓푣푒 7푠 푚푢푙푡푝푙푒푑 푡표푔푒푡ℎ푒푟 125 64 16807 1 Least Common Multiple The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. To find the LCM of some numbers: 1) express each of the numbers as a 50 = 2 ∙ 5 ∙ 5 42 = 2 ∙ 3 ∙ 7 36 = 2 ∙ 2 ∙ 3 ∙ 3 product of its prime factors, 2) re-write each of the factors using 50 = 21 ∙ 52 42 = 21 ∙ 31 ∙ 71 36 = 22 ∙ 32 exponents 3) for each base, select/circle the 50 = 21 ∙ ퟓퟐ 42 = 21 ∙ 31 ∙ ퟕퟏ 36 = ퟐퟐ ∙ ퟑퟐ largest power among all of the factors 4) multiply the selected powers 22 ∙ 32 ∙ 52 ∙ 71 = ퟔퟑퟎퟎ together Greatest Common Factor The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers. To find the GCF of some numbers: 1) express each of the numbers as a product 50 = 2 ∙ 5 ∙ 5 42 = 2 ∙ 3 ∙ 7 36 = 2 ∙ 2 ∙ 3 ∙ 3 of its prime factors, 2) re-write each of the factors using powers 50 = 21 ∙ 52 42 = 21 ∙ 31 ∙ 71 36 = 22 ∙ 32 3) for each base common to all factorizations, 50 = ퟐퟏ ∙ 52 42 = ퟐퟏ ∙ 31 ∙ 71 36 = ퟐퟐ ∙ 32 select/circle the smallest power among all ퟐퟏ 21 22 of the factors 4) multiply the selected powers together 21 = ퟐ Introduction to Integers Integers are an extension of whole numbers. They include whole numbers ({0,1,2,3, … }) and their opposites ({−1, −2, −3, … }). Two integers are opposites if they are each the same distance away from zero, but on opposite sides of zero. -7 0 7 7 and (−7) are opposites. Absolute Value The absolute value of a number may be thought of as its distance from zero. The absolute value of a number will be either positive or zero (non-negative). |17| = 17 |−21| = 21 |0| = 0 −|−3| = −3 |9 + 2| = |11| = 11 |23 − 23| = |0| = 0 −|14 − 19| = −|−5| = −5 |6 − 12| = |−6| = 6 2 Adding and Subtracting Integers The difference of two integers should be rewritten as the sum of the first integer and the opposite of the second. Then, follow the appropriate rule below for adding integers. The sum of two integers with the same sign is their The sum of two integers with different signs is their sum with their shared sign. difference, with the sign of the integer with the largest absolute value. −13 − 12 15 − (−7) −12 − (−9) −13 + (−12) −14 + (−3) 15 + 7 −3 + 13 −12 + 9 −11 + 2 |−3| < | 13| |−12| > |9| |−11| > |2| − (13 + 12) − (14 + 3) +(15 + 7) + (13 − 3) − (12 − 9) − (11 − 2) −25 −17 22 10 −3 −9 Multiplying and Dividing Integers The product or quotient of two integers with the The product or quotient of two integers with same sign is positive. different signs is negative. (−17)(−7) (14)(6) −65 ÷ (−13) 14 ÷ ( −7) (−4)(12) −33 ÷ 3 119 84 5 −2 −48 −11 Order of Operations The order of operations is a rule that lets us know which operations should be performed first. This ordering is as follows: Parentheses (Brackets and Braces) { [ ( ) ] } Exponents 42 3√27 √64 Multiplication and Division (from left to right) × 표푟 ÷ Addition and Subtraction (from left to right) + 표푟 − A group of numbers and operations inside of a parentheses will also follow the order of operations. Introduction to Fractions A fraction is another way of representing division. The bottom value of a fraction (the denominator) describes the number of equal parts that divide the whole. The top value (the numerator) describes how many of those parts there are. A proper fraction represents part of a whole and it has a value that is less than one and greater than zero. 푛푢푚푒푟푎푡표푟 2 5 푑푒푛표푚푛푎푡표푟 3 Equivalent Fractions Equivalent fractions have the same overall value, but the values of the numerators and denominators are different for each fraction. To make an equivalent fraction, we can either multiply or divide both the numerator and denominator of the fraction by the same number. 2 2 × 1 2 2 2 × 3 6 18 18 ÷ 1 18 18 18 ÷ 3 6 = = = = = = = = 5 5 × 1 5 5 5 × 3 15 36 36 ÷ 1 36 36 36 ÷ 3 12 2 2 × 5 10 2 2 × 10 20 18 18 ÷ 6 3 18 18 ÷ 18 1 = = = = = = = = 5 5 × 5 25 5 5 × 10 50 36 36 ÷ 6 6 36 36 ÷ 18 2 Simplifying Fractions Multiplying and Dividing Fractions In order to simplify a fraction, perform the prime Given two fractions, multiplication can be written factorization on the numerator and denominator in three different ways. of the fraction. Then, each common factor in both the numerator and denominator can be divided 3 1 3 1 3 1 ( ) ( ) × ∙ away. 10 4 10 4 10 4 60 135 3 3 3 42 162 40 40 40 ퟐ × 2 × ퟑ × 5 ퟑ × ퟑ × ퟑ × 5 ퟐ × ퟑ × 7 2 × ퟑ × ퟑ × ퟑ × 3 To multiply fractions, multiply the numerators ퟐ ퟑ 2 × 5 ퟑ ퟑ ퟑ 5 together and then multiply the denominators × × × × × ퟐ ퟑ 7 ퟑ ퟑ ퟑ 2 × 3 together. Then, simplify if possible. 2 × 5 5 1 × 1 × 1 × 1 × 1 × 3 3 2 1 3 4 7 3 7 2 × 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 7 3 6 4 10 9 7 2 × 5 5 3 × 3 2 × 1 3 × 4 7 × 3 7 2 × 3 5 × 7 3 × 6 4 × 10 9 × 7 10 5 9 2 12 21 7 6 35 18 40 63 1 3 1 Least Common Denominator 9 10 3 The Least Common Denominator (LCD) is the To divide fractions, swap the values of numerator smallest positive number that is a multiple of two and denominator (take the reciprocal) of the second or more denominators. To find the LCD of some fraction and then multiply the fractions. numbers: 5 6 1 7 3 7 28 = 2 ∙ 2 ∙ 7 ÷ ÷ ÷ 1) express each of the 6 7 2 8 4 10 denominators as a product 42 = 2 ∙ 3 ∙ 7 5 7 1 8 3 10 ( ) ( ) ( ) ( ) ( ) ( ) of its prime factors, 12 = 2 ∙ 2 ∙ 3 6 6 2 7 4 7 2 1 5 × 7 1 × 8 3 × 10 28 = 2 ∙ 7 6 × 6 2 × 7 4 × 7 2) re-write each of the 42 = 21 ∙ 31 ∙ 71 factors using powers 35 8 30 12 = 22 ∙ 31 36 14 28 3) for each base, 28 = ퟐퟐ ∙ ퟕퟏ 4 15 select/circle the largest 42 = 21 ∙ ퟑퟏ ∙ 71 7 14 power among all of the 2 1 factors 12 = 2 ∙ 3 4) multiply the selected 22 ∙ 31 ∙ 71 = ퟖퟒ powers together 4 Comparing Fractions To compare fractions, find the least common denominator (LCD) of the fractions and rewrite each fraction using that LCD. Then, the numerators can be compared. 3 7 10 1 3 4 , , , , 4 12 13 3 8 12 4 = ퟐퟐ 12 = 22 × ퟑퟏ 13 = ퟏퟑퟏ 3 = ퟑퟏ 8 = ퟐퟑ 12 = 22 × 31 ퟐퟐ × ퟑퟏ × ퟏퟑퟏ = ퟏퟓퟔ ퟐퟑ × ퟑퟏ = ퟐퟒ LCD(4,12,13) is ퟏퟓퟔ LCD(3,8,12) is ퟐퟒ 3 ퟑ ퟏퟑ 7 ퟏퟑ 10 ퟏퟐ 1 ퟖ 3 ퟑ 4 ퟐ ( ) ( ) , ( ) , ( ) ( ) , ( ) , ( ) 4 ퟑ ퟏퟑ 12 ퟏퟑ 13 ퟏퟐ 3 ퟖ 8 ퟑ 12 ퟐ 117 91 120 8 9 8 , , , , 156 156 156 24 24 24 Adding and Subtracting Fractions To add or subtract fractions with the same denominators, add or subtract the numerators and leave the denominators alone. 3 2 8 2 3 3 11 7 9 5 5 2 11 8 3 2 + − − + − + − + 5 5 11 11 5 5 15 15 4 4 13 13 12 12 7 7 5 6 0 18 4 7 3 5 5 11 5 15 4 13 12 7 1 0 6 1 1 5 4 To add or subtract fractions with different denominators, we will find a common denominator, and then change one or both of the fractions to have that common denominator.