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Divisibility and Factors

Divisibility and Factors. Lesson 4-1. Objectives: 1. to use divisibility tests 2. to find factors. Divisibility and Factors. Lesson 4-1. New Terms: 1. divisible – one integer is divisible by another if the remainder is 0 when you divide

andrew-gillespie
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Divisibility and Factors

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  1. Divisibility and Factors Lesson 4-1 Objectives: 1. to use divisibility tests 2. to find factors

  2. Divisibility and Factors Lesson 4-1 New Terms: 1. divisible – one integer is divisible by another if the remainder is 0 when you divide 2. factor – one integer is a factor of another integer (not zero) if it divides that integer with a remainder zero. Tips: the product of two integers is an integer, and both integers are factors of the product. Moreover, both integers divide the product, and the product is said to be divisible by each integer.

  3. Divisibility and Factors Lesson 4-1 Is the first number divisible by the second? a. 1,028 by 2 Yes; 1,028 ends in 8. b. 572 by 5 No; 572 doesn’t end in 0 or 5. c. 275 by 10 No; 275 doesn’t end in 0.

  4. Divisibility and Factors Lesson 4-1 Is the first number divisible by the second? a. 1,028 by 3 No; 1 + 0 + 2 + 8 = 11; 11 is not divisible by 3. b. 522 by 9 Yes; 5 + 2 + 2 = 9; 9 is divisible by 9.

  5. Divisibility and Factors Lesson 4-1 Ms. Washington’s class is having a class photo taken. Each row must have the same number of students. There are 35 students in the class. How can Ms. Washington arrange the students in rows if there must be at least 5 students, but no more than 10 students, in each row? 1 • 35, 5 • 7  Find pairs of factors of 35. There can be 5 rows of 7 students, or 7 rows of 5 students.

  6. Exponents Lesson 4-2 Objectives: 1. to use exponents 2. to use the order of operations with exponents

  7. Exponents Lesson 4-2 New Terms: 1. exponents – you can use exponents to show repeated multiplication 2. power – a power has two parts, a base and an exponent Tips: the exponent is placed to the upper right of the base, and it only applies to that base. If an exponent has as its base an expression, that expression must be written in parentheses.

  8. (–11)4 Include the negative sign within parentheses. –5 • x • x • x • y • y Rewrite the expression using the Commutative and Associative Properties. –5x3y2 Write x • x • x and y • y using exponents. Exponents Lesson 4-2 Write using exponents. a. (–11)(–11)(–11)(–11) b. –5 • x • x • y • y • x

  9. 104 = 10 • 10 • 10 • 10 The exponent indicates that the base 10 is used as a factor 4 times. = 10,000 light-years Multiply. Exponents Lesson 4-2 Suppose a certain star is 104 light-years from Earth. How many light-years is that? The star is 10,000 light-years from Earth.

  10. 3(1 + 4)3 = 3(5)3 Work within parentheses first. = 3 • 125 Simplify 53. = 375 Multiply. 7(w + 3)3 + z = 7(–5 + 3)3 + 6 Replace w with –5 and z with 6. = 7(–2)3+ 6 Work within parentheses. = 7(–8) + 6 Simplify (–2)3. = –56 + 6 Multiply from left to right. = –50 Add. Exponents Lesson 4-2 a. Simplify 3(1 + 4)3. b. Evaluate 7(w + 3)3 + z, for w = –5 and z = 6.

  11. Prime Factorization and Greatest Common Factor Lesson 4-3 Objectives: 1. to find the prime factorization of a number 2. to find the greatest common factor (GCF) of two or more numbers New Terms: 1. prime number – is an integer greater than 1 with exactly two positive factors, 1 and itself 2. composite number – is an integer greater than 1 with more than two positive factors 3. greatest common factor – factors that are the same for two or more numbers or expressions are common factors. The greatest of these is called the GCF. Tips: remember a factor is a number that divides evenly into another number with a remainder of zero.

  12. Prime Factorization and Greatest Common Factor Lesson 4-3 State whether each number is prime or composite. Explain. a. 46 Composite; 46 has more than two factors, 1, 2, 23, and 46. b. 13 Prime; 13 has exactly 2 factors, 1 and 13.

  13. Prime 3 • 91 Start with a prime factor. Continue branching. Prime 7 • 13 Stop when all factors are prime. 3 • 7 • 13 Write the prime factorization. Prime Factorization and Greatest Common Factor Lesson 4-3 Use a factor tree to write the prime factorization of 273. 273 273 = 3 • 7 • 13 

  14. 24 = 23 • 3 Write the prime factorizations. 30 = 2 • 3 • 5 Find the common factors. GCF = 2 • 3 Use the lesser power of the common factors. = 6 36ab2 = 22 • 32 • a • b2 Write the prime factorizations. Find the common factors. 81b = 34 • b GCF = 32 • b Use the lesser power of the common factors. = 9b Prime Factorization and Greatest Common Factor Lesson 4-3 Find the GCF of each pair of numbers or expressions. a. 24 and 30 The GCF of 24 and 30 is 6. b. 36ab2 and 81b The GCF of 36ab2 and 81b is 9b.

  15. Simplifying Fractions Lesson 4-4 Objectives: 1. to find equivalent fractions 2. to write fractions in simplest for New Terms: 1. simplest form – a fraction is in simplest form when the numerator and the denominator have no common factors other than 1. Tips: always check your answers and the steps involved in finding the answers

  16. 18 • 2 21 • 2 = 36 42 = 18 ÷ 3 21 ÷ 3 = 6 7 = 6 7 36 42 18 21 The fractions and are both equivalent to . Simplifying Fractions Lesson 4-4 18 21 Find two fractions equivalent to . 18 21 a. 18 21 b.

  17. 21 28 21 28 21 ÷ 7 28 ÷ 7 = Divide the numerator and denominator by the GCF, 7. 3 4 = Simplify. 3 4 of the students in the class buy their lunches in the cafeteria. Simplifying Fractions Lesson 4-4 You learn that 21 out of the 28 students in a class, or , buy their lunches in the cafeteria. Write this fraction in simplest form. The GCF of 21 and 28 is 7.

  18. Divide the numerator and denominator by the common factor, p. p 2p p1 2p1 = 1 2 Simplify. = Simplifying Fractions Lesson 4-4 Write in simplest form. p 2p a.

  19. Write as a product of prime factors. 14q2rs3 8qrs2 2 • 7 • q • q • r • s • s • s 2 • 2 • 2 • q • r • s • s = Divide the numerator and denominator by the common factors. 21 • 7 • q1 • q • r1 • s1 • s1 • s 21 • 2 • 2 • q1 • r1 • s1 • s1 = 7 • q • s 2 • 2 Simplify. = 7 • q • s 4 = Simplify. 7qs 4 = Simplifying Fractions Lesson 4-4 (continued) 14q2rs3 8qrs2 b.

  20. Problem Solving Strategy: Solve a Simpler Problem Lesson 4-5 Additional Examples Aaron, Chris, Maria, Sonia, and Ling are on a class committee. They want to choose two members to present their conclusions to the class. How many different groups of two members can they form?

  21. First, pair Aaron with each of the four other committee members. Next, pair Chris with each of the three members left. Since Aaron and Chris have already been paired, you don’t need to count them again. Repeat for the rest of the committee members. Chris Maria Maria Aaron Chris Sonia Sonia Ling Ling Sonia Maria Sonia Ling Ling Problem Solving Strategy: Solve a Simpler Problem Lesson 4-5 Additional Examples (continued) Each successive tree has one less branch. There are 10 different groups of two committee members.

  22. Rational Numbers Lesson 4-6 Objectives: 1. to identify and graph rational numbers 2. to evaluate fractions containing variables New Terms: 1. rational number – is any number you can write as a fraction Tips: the quotient of two integers with the same sign is positive

  23. 2 3 4 6 6 9 = = = … Numerators and denominators are positive. 2 3 –2 –3 –4 –6 = = = … Numerators and denominators are negative. Rational Numbers Lesson 4-6 2 3 Write two lists of fractions equivalent to .

  24. Rational Numbers Lesson 4-6 Additional Examples Graph each rational number on a number line. 3 4 a. – b. 0.5 c. 0 1 3 d.

  25. f – i t a = Use the acceleration formula. 90 – 0 5 = Substitute. 90 5 = Subtract. = 18 Write in simplest form. Rational Numbers Lesson 4-6 A fast sports car can accelerate from a stop to 90 ft/s in 5 seconds. What is its acceleration in feet per second per second (ft/s2)? Use the formula a = , where a is acceleration, f is final speed, i is initial speed, and t is time. f – i t The car’s acceleration is 18 ft/s2.

  26. Exponents and Multiplication Lesson 4-7 Objectives: 1. to multiply powers with the same base 2. to find a power of a power

  27. Exponents and Multiplication Lesson 4-7 Tips: when in doubt, write it out.

  28. 52 • 53 = 52+ 3 Add the exponents of powers with the same base. = 3,125 Simplify. x5 • x7 • y2 • y = x5 + 7 • y2+1 Add the exponents of powers with the same base. = x12y3 Simplify. Exponents and Multiplication Lesson 4-7 Simplify each expression. a. 52 • 53 = 55 b. x5 • x7 • y2 • y

  29. 3a3 • (–5a4) = 3 • (–5) • a3 • a4 Use the Commutative Property of Multiplication. = –15a3+ 4 Add the exponents. = –15a7 Simplify. Exponents and Multiplication Lesson 4-7 Simplify 3a3 • (–5a4).

  30. (23)3 = (2)3• 3 Multiply the exponents. = (2)9 Simplify the exponent. = 512 Simplify. (g5)4 = g5• 4 Multiply the exponents. = g20 Simplify the exponent. Exponents and Multiplication Lesson 4-7 Simplify each expression. a. (23)3 b. (g5)4

  31. Exponents and Division Lesson 4-8 Objectives: 1. To divide expressions containing exponents 2. To simplify expressions with integer exponents

  32. Exponents and Division Lesson 4-8 Tips: Sometimes students think an expression with a negative exponent makes the expression negative, that is not true. A negative exponent makes the number smaller (unless the base is less than one but greater than zero).

  33. 412 48 412 48 = 412– 8 Subtract the exponents. = 44 Simplify the exponent. = 256 Simplify. w18 w13 = w18– 13 Subtract the exponents. = w5 Simplify the exponent. Exponents and Division Lesson 4-8 Simplify each expression. a. w18 w13 b.

  34. (–12)73 (–12)73 = (–12)73– 73 Subtract the exponents. = (–12)0 Simplify. 1 4 8 32 8s20 32s20 = s0 Subtract the exponents. Simplify . 1 4 = • 1 Simplify s0. 1 4 = Multiply. Exponents and Division Lesson 4-8 Simplify each expression. (–12)73 (–12)73 a. = 1 8s20 32s20 b.

  35. 612 614 = 612– 14 Subtract the exponents. = 6–2 1 62 = Write with a positive exponent. 1 36 = Simplify. z4 z15 = z4– 15 Subtract the exponents. = z–11 1 z11 = Write with a positive exponent. Exponents and Division Lesson 4-8 Simplify each expression. 612 614 a. z4 z15 b.

  36. a2b3 ab15 = a2 –1b3– 15  Use the rule for Dividing Powers with the Same Base. = ab–12 Subtract the exponents. Exponents and Division Lesson 4-8 a2b3 ab15 Write without a fraction bar.

  37. Scientific Notation Lesson 4-9 Objectives: 1. to write and evaluate numbers in scientific notation 2. to calculate with scientific notation New Terms: 1. scientific notation– a way to write numbers using powers of 10 2. standard notation– write the number using all digit and decimal places Tips: be careful with which way you move the decimal, think if the number is suppose to be smaller or larger

  38. Move the decimal point to get a decimal greater than 1 but less than 10. 6 places 6.3 Drop the zeros after the 3. 6.3 x 106 You moved the decimal point 6 places. The original number is greater than 10. Use 6 as the exponent of 10. Scientific Notation Lesson 4-9 About 6,300,000 people visited the Eiffel Tower in the year 2000. Write this number in scientific notation. 6,300,000

  39. Move the decimal point to get a decimal greater than 1 but less than 10. 4 places 3.7 Drop the zeros before the 3. 3.7 x 10–4 You moved the decimal point 4 places. The original number is less than 1. Use –4 as the exponent of 10. Scientific Notation Lesson 4-9 Write 0.00037 in scientific notation. 0.00037

  40. Write zeros while moving the decimal point. 36,000 Rewrite in standard notation. Write zeros while moving the decimal point. 0.0072 Rewrite in standard notation. Scientific Notation Lesson 4-9 Write each number in standard notation. a. 3.6 x 104 3.6000 b. 7.2 x 10–3 007.2

  41. 0.107x 1012 = 1.07 x 10–1x 1012 Write 0.107 as 1.07  10–1. = 1.07 x 1011 Add the exponents. Write 515.2 as 5.152  102. 515.2x 10–4 = 5.152 x 102x 10–4 = 5.152 x 10–2 Add the exponents. Scientific Notation Lesson 4-9 Write each number in scientific notation. a. 0.107 x 1012 b. 515.2 x 10–4

  42. Write each number in scientific notation. 0.035 x 104 710 x 10–1 0.69 x 102 3.5 x 102 7.1 x 10 6.9 x 10 Order the powers of 10. Arrange the decimals with the same power of 10 in order. 6.9 x10 7.1 x10 3.5 x102 Write the original numbers in order. 0.69 x 102, 710 x 10–1, 0.035 x 104 Scientific Notation Lesson 4-9 Order 0.035 x 104, 710 x 10–1, and 0.69 x 102 from least to greatest.

  43. (4 x 10–6)(7 x 109) = 4 x 7 x 10–6x 109 Use the Commutative Property of Multiplication. = 28 x 10–6x 109 Multiply 4 and 7. = 28x 103 Add the exponents. Write 28 as 2.8  101. = 2.8 x 101x 103 = 2.8 x 104 Add the exponents. Scientific Notation Lesson 4-9 Multiply 4 x 10–6 and 7 x 109. Express the result in scientific notation.

  44. (6.02 x 1023)(1.67 x 10–27) Multiply number of atoms by weight of each. = 6.02 x 1.67 x 1023x 10–27 Use the Commutative Property of Multiplication. 10.1 x 1023x 10–27 Multiply 6.02 and 1.67. = 10.1x 10–4 Add the exponents. = 1.01 x 101x 10–4 Write 10.1 as 1.01  101. = 1.01 x 10–3 Add the exponents. Scientific Notation Lesson 4-9 In chemistry, one mole of any element contains approximately 6.02 x 1023 atoms. If each hydrogen atom weighs approximately 1.67 x 10–27 kg, approximately how much does one mole of hydrogen atoms weigh? One mole of hydrogen atoms weighs approximately 1.01 x 10–3 kg.

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