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Divisibility and Factors

Learn about divisibility rules, factor pairs, prime factorization, and common factors in this comprehensive pre-algebra lesson. Practice with various exercises and quick checks to enhance your understanding.

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Divisibility and Factors

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  1. Divisibility and Factors PRE-ALGEBRA LESSON 4-1 Karen placed 56 bottles into boxes that each held 6 bottles. How many boxes did she use? 10 4-1

  2. 354 2 354 3 Divisibility and Factors PRE-ALGEBRA LESSON 4-1 (For help, go to Skills Handbook p. 760.) Find each quotient. 1. 480 ÷ 3 2. 365 ÷ 5 3. 459 ÷ 9 4. 288 ÷ 6 5.6. Check Skills You’ll Need 4-1

  3. Solutions 1. 160 3 480 –3 18 –18 0 73 5 365 –35 15 –15 0 2. 3. 51 9 459 –45 9 – 9 0 Divisibility and Factors PRE-ALGEBRA LESSON 4-1 4-1

  4. 4. 48 6 288 –24 48 –48 0 5. 177 2 354 –2 15 –14 14 –14 0 6. 118 3 354 –3 5 –3 24 –24 0 Divisibility and Factors PRE-ALGEBRA LESSON 4-1 Solutions (continued) 4-1

  5. Divisibility and Factors PRE-ALGEBRA 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. Quick Check 4-1

  6. Divisibility and Factors PRE-ALGEBRA 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. Quick Check 4-1

  7. Divisibility and Factors PRE-ALGEBRA 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. Quick Check 4-1

  8. Divisibility and Factors PRE-ALGEBRA LESSON 4-1 State whether each number is divisible by 2, 3, 5, 9, or 10. 1. 18 2. 90 3. 81 4. 25 5. List the positive factors of 36. 2, 3, 9 2, 3, 5, 9, 10 3, 9 5 1, 2, 3, 4, 6, 9, 12, 18, 36 4-1

  9. Exponents PRE-ALGEBRA LESSON 4-2 Which word best completes the statement: sometimes, always, or never? If the product of two factors is zero, both factors are zero. sometimes 4-2

  10. Exponents PRE-ALGEBRA LESSON 4-2 (For help, go to Lesson 1-9.) Find each product. 1. 3 • 3 • 3 • 3 2. –12 • (–12) 3. (–4)(–4)(–4) 4. 10 • 10 • 10 • 10 Check Skills You’ll Need 4-2

  11. Exponents PRE-ALGEBRA LESSON 4-2 Solutions 1. 3 • 3 • 3 • 3 = 81 2. –12 • (–12) = 144 3. (–4)(–4)(–4) = –64 4. 10 • 10 • 10 • 10 = 10,000 4-2

  12. (–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 PRE-ALGEBRA LESSON 4-2 Write using exponents. a. (–11)(–11)(–11)(–11) b. –5 • x • x • y • y • x Quick Check 4-2

  13. 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 PRE-ALGEBRA LESSON 4-2 Suppose a certain star is 104 light-years from Earth. How many light-years is that? Quick Check 4-2

  14. 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 PRE-ALGEBRA LESSON 4-2 a. Simplify 3(1 + 4)3. b. Evaluate 7(w + 3)3 + z, for w = –5 and z = 6. Quick Check 4-2

  15. Exponents PRE-ALGEBRA LESSON 4-2 Write using exponents. 1.x • y • z • x • z2.a • b • b • b • 3 3. Simplify 5(2 + 4)2. 4. Evaluate (g3 – 7)2 • 5 + 4, for g = 3. x2yz2 3ab3 180 2,004 4-2

  16. Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4-3 Find three integers whose sum is 12 and whose product is 42. 2, 3, 7 4-3

  17. 4-3 Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4-3 (For help, go to Lesson 4-1.) List the positive factors of each number. 1. 15 2. 35 3. 7 4. 20 5. 100 6. 121 Check Skills You’ll Need

  18. 4-3 Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4-3 Solutions 1. 1 • 15, 3 • 5; 1, 3, 5, 15 2. 1 • 35, 5 • 7; 1, 5, 7, 35 3. 1 • 7; 1, 7 4. 1 • 20, 2 • 10, 4 • 5; 1, 2, 4, 5, 10, 20 5. 1 • 100, 2 • 50, 4 • 25, 5 • 20, 10 • 10; 1, 2, 4, 5, 10, 20, 25, 50, 100 6. 1 • 121, 11 • 11; 1, 11, 121

  19. 4-3 Prime Factorization and Greatest Common Factor PRE-ALGEBRA 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. Quick Check

  20. 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. 4-3 Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4-3 Use a factor tree to write the prime factorization of 273. 273 273 = 3 • 7 • 13  Quick Check

  21. 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 4-3 Prime Factorization and Greatest Common Factor PRE-ALGEBRA 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. Quick Check

  22. 4-3 Prime Factorization and Greatest Common Factor PRE-ALGEBRA LESSON 4-3 Tell whether each number is prime or composite. 1. 123 2. 47 Write the prime factorization for each number. 3. 64 4. 45 Find the GCF for each pair. 5. 80 and 120 6. 62b3c2d and 31b2c3d composite prime 26 32 • 5 40 31b2c2d

  23. 4-4 Simplifying Fractions PRE-ALGEBRA LESSON 4-4 Find the GCF for each pair of numbers. a. 12 and 18 b. 15 and 60 c. 54 and 60 6 15 6

  24. 4-4 Simplifying Fractions PRE-ALGEBRA LESSON 4-4 (For help, go to Lesson 4-3.) Find each GCF. 1. 14, 21 2. 48, 60 3. 5mn, 15m2n4. 63r2, 48s3 Check Skills You’ll Need

  25. 4-4 Simplifying Fractions PRE-ALGEBRA LESSON 4-4 Solutions 1. 14, 21 14 = 2 • 7 21 = 3 • 7 GCF = 7 2. 48, 60 48 = 24 • 3 60 = 22 • 3 • 5 GCF = 22 • 3 = 12 3. 5mn, 15m2n 5mn = 5 • m • n 15m2n = 3 • 5 • m2 • n GCF = 5 • m • n = 5mn 4. 63r2, 48s3 63r2 = 32 • 7 • r2 48s3 = 24 • 3 • s3 GCF = 3

  26. 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 . 4-4 Simplifying Fractions PRE-ALGEBRA LESSON 4-4 18 21 Find two fractions equivalent to . 18 21 a. 18 21 b. Quick Check

  27. 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. 4-4 Simplifying Fractions PRE-ALGEBRA 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. Quick Check

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

  29. 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 = 4-4 Simplifying Fractions PRE-ALGEBRA LESSON 4-4 (continued) 14q2rs3 8qrs2 b. Quick Check

  30. Sample answer: Sample answer: and wx4y8 w2x3y xy7 w 10 30 22 32 33 48 1 3 1 4 and 11 16 7 21 13 52 4-4 Simplifying Fractions PRE-ALGEBRA LESSON 4-4 Find two fractions equivalent to each fraction. 1.2. Write in simplest form. 3.4.

  31. 1 10 9 10 5 8 3 8 7 8 1 8 5 6 3 4 4-5 Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4-5 Choose the symbol <, =,or >that makes each statement true. a. 2 + 1 + 2 ? 4 + 1 b. 3 + 2 +?4 + 3 = <

  32. 4-5 Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4-5 (For help, go to Skills Handbook p. 775.) Compare. Use > to < to complete each statement. 1. 3 0 2. –16 –25 3. 0 1 4. –30 –20 Check Skills You’ll Need

  33. 4-5 Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4-5 (For help, go to Skills Handbook p.775.) Solutions 1. 3 > 0 2. –16 > –25 3. 0 < 1 4. –30 < –20

  34. 4-5 Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4-5 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?

  35. 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 4-5 Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4-5 (continued) Each successive tree has one less branch. Quick Check There are 10 different groups of two committee members.

  36. 4-5 Problem Solving Strategy: Solve a Simpler Problem PRE-ALGEBRA LESSON 4-5 Solve each problem. 1. Twelve people are at a party. Each person greets each of the other persons exactly once. How many greetings will there be in all? 2. How many different pairs of classmates can you choose from six classmates? 3. Each small box is a square. What is the number of different squares shown? 66 15 pairs 17

  37. 4 7 4-6 Rational Numbers PRE-ALGEBRA LESSON 4-6 29,716 52,003 Write in simplest form.

  38. 6 8 2 10 14 21 28 35 4-6 Rational Numbers PRE-ALGEBRA LESSON 4-6 (For help, go to Lesson 4-4.) Write in simplest form. 1. 2. 3. 4. Check Skills You’ll Need

  39. = = = = = = = = 2 ÷ 2 10 ÷ 2 28 ÷ 7 35 ÷ 7 14 ÷ 7 21 ÷ 7 6 ÷ 2 8 ÷ 2 3 4 4 5 2 3 1 5 6 8 28 35 2 10 14 21 4-6 Rational Numbers PRE-ALGEBRA LESSON 4-6 Solutions 1. 2. 3. 4.

  40. 2 3 4 6 6 9 = = = … Numerators and denominators are positive. 2 3 –2 –3 –4 –6 = = = … Numerators and denominators are negative. 4-6 Rational Numbers PRE-ALGEBRA LESSON 4-6 2 3 Write two lists of fractions equivalent to . Quick Check

  41. 4-6 Rational Numbers PRE-ALGEBRA LESSON 4-6 Graph each rational number on a number line. 3 4 a. – b. 0.5 c. 0 1 3 d. Quick Check

  42. f – i t a = Use the acceleration formula. 90 – 0 5 = Substitute. 90 5 = Subtract. = 18 Write in simplest form. 4-6 Rational Numbers PRE-ALGEBRA 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. Quick Check

  43. –5 –6 10 12 15 18 Sample: , , 5 6 1 4 1 2 4-6 Rational Numbers PRE-ALGEBRA LESSON 4-6 Write three fractions equivalent to the given fraction. 1. Graph each rational number on a number line. 2. a.b. – c. 1.5 d. 0.4 3. A car can accelerate from 0 to 70 ft/s in 5 s. What is the acceleration of the car in feet per second per second (ft/s2)? 14 ft/s2

  44. 4-7 Exponents and Multiplication PRE-ALGEBRA LESSON 4-7 23 32 is the prime factorization for __. ? 72

  45. 4-7 Exponents and Multiplication PRE-ALGEBRA LESSON 4-7 (For help, go to Lesson 4-2.) Write using exponents. 1.k • k • k • k2.m • n • m • n 3. 2 • 2 • 2 • 2 4. 5 • 5 • 5 Check Skills You’ll Need

  46. 4-7 Exponents and Multiplication PRE-ALGEBRA LESSON 4-7 Solutions 1.k • k • k • k = k42.m • n • m • n = m • m • n • n = m2n2 3. 2 • 2 • 2 • 2 = 244. 5 • 5 • 5 = 53

  47. 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. 4-7 Exponents and Multiplication PRE-ALGEBRA LESSON 4-7 Simplify each expression. a. 52 • 53 = 55 b. x5 • x7 • y2 • y Quick Check

  48. 3a3 • (–5a4) = 3 • (–5) • a3 • a4 Use the Commutative Property of Multiplication. = –15a3+ 4 Add the exponents. = –15a7 Simplify. 4-7 Exponents and Multiplication PRE-ALGEBRA LESSON 4-7 Simplify 3a3 • (–5a4). Quick Check

  49. (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. 4-7 Exponents and Multiplication PRE-ALGEBRA LESSON 4-7 Simplify each expression. a. (23)3 b. (g5)4 Quick Check

  50. 28r s6 9 4-7 Exponents and Multiplication PRE-ALGEBRA LESSON 4-7 Simplify each expression. 1. 22 • 232.g2 • h2 • h4 • h 3. 4r s • 7r s54. –(22)5 5. (v3)8 32 g2 • h7 3 6 –1,024 v24

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