Math 365 Lecture Notes S. Nite 8/18/2012 Page 1 of 9 Integers and Number Theory Exam 2 Review Chapters 4-5 Divisibility Theorem 4-1 If d a, n I, then d (a n) Theorem 4-2 If d a, and d b, then d (a+b). If d a, and d does not b, then d does not (a+b). If d a, and d b, then d (a-b). If d a, and d does not b, then d does not (a-b). Divisibility Rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11 Prime Numbers and Composite Numbers Prime Factorization Fundamental Theorem of Arithmetic: Each composite number can be written as a product of primes in one, and only one, way (except for order) Number of Divisors Fundamental Counting Principle Theorem 4-4: If d is a divisor or n, then n/d is also a divisor of n. Theorem 4-5: If n is composite, then n has a prime factor p such that p 2 n. Theorem 4-6: If n > 1 and not divisible by any prime such that p 2 n, then n is prime. Creating a Sieve of Prime Numbers Greatest Common Factor/Divisor (GCF or GCD) Colored Rods, Intersection of Sets, Prime Factorization Relatively Prime GCF(0, a) = a Euclidean Algorithm Theorem 4-7: If a b, a, b > 0, then GCF (a, b) = GCF (r, b) where r is the remainder when a is divided by b. Least Common Multiple (LCM) Colored Rods, Intersection of Sets, Prime Factorization Theorem 4-8: If a, b N, then GCF(a, b) LCM(a, b) = a b.
Math 365 Lecture Notes S. Nite 8/18/2012 Page 2 of 9 Division by Primes Method Natural Numbers, Whole Numbers, and Integers (negative integers are opposites of positive integers or natural numbers) Integer Addition Models Chip, Charged field, Patterns, Number Line Absolute Value Integer Addition Properties Closure, Commutative, Associative, Identity element, Additive inverse Integer Subtraction Models Chip, Charged field, Patterns, Number Line Integer Subtraction Definition; Closure Order of Operations Integer Multiplication Models Chip, Charged field, Patterns, Number Line Integer Multiplication Properties Closure, Commutative, Associative, Identity element, Zero multiplication Distributive Property of Multiplication over Addition Difference of Squares Division of Integers Definition of Less Than Extending the Coordinate System (x, y) Rational Numbers as Fractions Rational Number; Numerator; Denominator; Fraction; Proper Fraction; Improper Fraction a a an Fundamental Law of Fractions: is any fraction and n 0, then = b b bn Ways to Use Fractions Division problem; Part of a whole; Ratio; Probability Equivalent Fractions (using LCD) Using Equivalent Fractions and prime factoring to Simplify Fractions Using Equivalent Fractions to Order Fractions Properties and Theorems: Denseness Property
Math 365 Lecture Notes S. Nite 8/18/2012 Page 3 of 9 Adding/Subtracting Fractions with and without fraction bars Adding/Subtracting Mixed Numbers with and without fraction bars a c a c a c Multiplying Fractions If and are any rational numbers, then = b d b d b d Properties o Multiplicative identity of rational numbers o Multiplicative inverse of rational numbers o Distributive Property of multiplication over addition for rational numbers o Multiplication property of equality for rational numbers o Multiplication property of zero for rational numbers Multiplying Fractions area model; commutative property; with/without fraction bars Multiplication with Mixed Numbers distributive, changing to an improper fraction Definition of Division of Rational Numbers If c e = d f a c c a, R, and 0, then b d d b e c e a if, and only if, is the unique rational number such that = f d f b Algorithm for Division of Fractions b a d c = b a c d where d c 0 Dividing Fraction with and without fraction bars Mental Math Strategies Rules of Exponents
Math 365 Lecture Notes S. Nite 8/18/2012 Page 4 of 9 Review Problems 1. Test 3,201,012 for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11 2. Without dividing, how can you tell whether 24,013 is divisible by 12? 3. Jennifer claims that a number is divisible by 4 if each of the last two digits is divisible by 4. Is she correct? Explain. 4. Explain why 3 2 5 3 9 2 is not a prime factorization. Write the prime factorization. 5. Use the prime factorization method to find the GCD of 12,870 and 11,165. 6. Use the colored rods method to model finding the LCM of 6 and 9.
Math 365 Lecture Notes S. Nite 8/18/2012 Page 5 of 9 7. Use the number line model to find 7 + -4. 8. Does x represent a negative number? Explain. 9. Use the charged field model to find -6 4. 10. Use the pattern model to find (-4) (-2). 11. Identify the property illustrated: a. (-4)(-7) I b. (-9)[5 + (-8)] = (-9) 5 + (-9)(-8) 12. Use the difference-of-squares formula to simplify: (5-100)(5 + 100) 13. Factor: mnp + mn m. 14. Use the distributive property to compute 997 2.
Math 365 Lecture Notes S. Nite 8/18/2012 Page 6 of 9 15. Annie used the charged-field model to show that -3(-4) = 12. She said that this proves that the product of two negative numbers is a positive number. How do you respond? 1 16. If the given model represents 5, draw a model that represents the whole. 1 17. Jan noticed that 5 > 4, but 5 1 < 4. She wants to know if the same holds true for negative numbers and why or why not. How do you respond? 18. Arrange the following fractions from least to greatest. 5 22 6 25 9 55 19. Suppose a large pizza is divided into 12 equal size pieces and a 1 1 + medium pizza is divided into 8 equal size pieces. Does 12 8 represent the amount that you received? Explain.
Math 365 Lecture Notes S. Nite 8/18/2012 Page 7 of 9 b 20. The sum of two fractions is 1. If one of the fractions is a, what is the other fraction? 4 33 7 15 21. Compute and simplify: 22 + 9 2 1. 22. Use a rectangular region to model 3 5 23 x = 19 69 23. Solve for x: 76 24. A student claims that division always makes things smaller so 2 1 5 cannot be 10 because that is greater than the number she started with. What is an appropriate response?
Math 365 Lecture Notes S. Nite 8/18/2012 Page 8 of 9 173 25. Can 625 be written as a terminating decimal? Explain how you know. 26. Round 825.49 as specified. a. to the nearest tenth b. to the nearest unit c. to the nearest ten d. to the nearest hundred 2 4 3 4 7 27. Multiply using the distributive property: 5 28. Order the following numbers from least to greatest. 2.72 2.727 2. 7
Math 365 Lecture Notes S. Nite 8/18/2012 Page 9 of 9 2 5 29. Model with fraction bars: 10 3 23 30. Find a fraction between 55 24 and 55 20 28 31. 63 213