Integer Multiplication and Division

• These notes cover integer multiplication and division, including rules for signs and evaluating expressions.

True or False Statements

• The product of a positive and a negative number is negative (True).
• The product of an even number of negative numbers is negative (False).
• Dividing two negative integers results in a negative number (False).
• The quotient of a positive integer divided by a negative integer is positive (False).
• The order of integers does not affect the product when multiplying (True).
• Dividing an integer by 1 results in the original integer (True).

Determining Positive or Negative Expressions (Multiplication)

• Multiplying a negative number by a positive number yields a negative result.
• Multiplying a positive number by a negative number yields a negative result.
• Multiplying two negative numbers yields a positive result.

Determining Positive or Negative Expressions (Division)

• Dividing a negative number by a positive number yields a negative result.
• Dividing a negative number by a negative number yields a positive result.
• Dividing a positive number by a negative number yields a negative result.

Evaluating Expressions Based on 6 x 4 = 24

• (-6) x 4 = -24
• 6 x (-4) = -24
• (-6) x (-4) = 24

Evaluating Expressions Based on 3 x 2 x 5 = 30

• 3 x (-2) x 5 = -30
• (-3) x 2 x (-5) = 30
• (-3) x (-2) x 5 = 30
• (-3) x (-2) x (-5) = -30

Evaluating Expressions Based on 42 ÷ 7 = 6

• (-42) ÷ 7 = -6
• 42 ÷ (-7) = -6
• (-42) ÷ (-7) = 6

Evaluating Expressions Based on 924 ÷ 28 = 33

• (-924) ÷ 28 = -33
• 924 ÷ (-28) = -33
• (-924) ÷ (-28) = 33

Evaluating Multiplication Expressions

• 6 x (-6) = -36
• (-12) x (-10) = 120
• (-9) x 0 = 0
• (-4) x (-4) x (-4) = -64
• (-4) x (-11) = 44
• (-4)² = 16
• (-12)² = 144
• 15 x (-3) = -45
• (-9) x 1 = -9
• (-11) x 4 = -44
• (-4) x (-4) = 16
• 4 x (-7) x 5 = -140
• (-4)³ = -64
• -(-12)² = -144
• -(-8)³ = 512
• (-5)³ = -125
• (-3) x 4 x (-3) = 36
• (-6) x (-8) x (-3) = -144
• -4 x 4 x 5 = -80
• (-8) x (-5) x (-3) x 6 = -720

Completing Multiplication Equations

• -11 x (-5) = 55
• -6 x 11 = -66
• -7 x 8 = -56
• 7 x -12 = -84

Solving Multiplication Equations

• (-11) x r = -66, r = 6
• n x (-3) = -36, n = 12
• 7 x q = -28, q = -4
• t x 7 = -42, t = -6

Finding the Value of 'n' in Multiplication Statements

• n x (-3) = 21, n = -7
• n x (-10) = -40, n = 4

Evaluating Division Expressions

• (-2) ÷ (-2) = 1
• 12 ÷ (-4) = -3
• -11 ÷ -11 = 1
• (-2) ÷ 1 = -2
• 4 ÷ (-1) = -4
• 0 ÷ (-36) = 0
• -5 ÷ 5 = -1
• 10 ÷ -1 = -10
• 42 ÷ -7 = -6
• (-30) ÷ 6 = -5
• 21 ÷ (-7) = -3
• (-50) ÷ (-10) = 5
• (-72) ÷ (-9) = 8
• -48 ÷ -12 = 4
• (-22) ÷ 11 = -2
• -60 ÷ 10 = -6

Completing Division Equations

• 54 ÷ (-6) = -9
• -35 ÷ 5 = -7
• -55 ÷ 5 = -11
• (-72) ÷ -8 = 9

Solving Division Equations

• (-108) ÷ n = -12, n = 9
• p ÷ (-8) = 11, p = -88
• (-48) ÷ t = 6, t = -8
• k ÷ 8 = -6, k = -48

Finding the Value of 'n' in Division Statements

• n ÷ 3 = -9, n = -27
• n ÷ (-5) = -4, n = 20

Evaluating Combined Multiplication and Division Expressions

• (-4) x 3 ÷ (-2) = 6
• 5 x (-3) x (-7) = 105
• 10 x 8 ÷ (-5) = -16
• (-6) x (-4) x (-3) = -72
• (-14) x (-4) ÷ 7 = 8
• 8 x (-15) ÷ 4 = -30
• (18 ÷ (-3)) ÷ (-2) = 3
• (-126) ÷ ((-6) x (-3)) = -7

Reasoning About Integer Operations

• There is no real number solution to x² = -1 because the square of any real number is non-negative.
• The product of two integers can be zero if at least one of the integers is zero.
• -4 x 8 x (-2) = 32 and -4 x (-2) x 8 = 32, the answers are the same due to the commutative property of multiplication.
• (9 x -6) x 3 and 9 x (-6 x 3) yield the same result due to the associative property of multiplication.

Patterns in Multiplying Negative Terms

• Product Number of negative terms Result
• −1 × −1 2 1
• −1 × −1 × −1 3 –1
• −1 × −1 × −1 × −1 4 1
• −1 × −1 × −1 × −1 × −1 5 –1
• −1 × −1 × −1 × −1 × −1 × −1 6 1
• The number of negative terms determines the sign of the result based on whether it is even (positive) or odd (negative).

Problem Solving with Integers

• Two integers with a product of -35 and a sum of -2: -7 and 5

• For xy > 0, (x, y) must lie in quadrants I or III.

• For xy < 0, (x, y) must lie in quadrants II or IV.

• Easy Questions

  1. 5×3=5 \times 3 =5×3=
  2. −4×2=-4 \times 2 =−4×2=
  3. −6×−3=-6 \times -3 =−6×−3=
  4. 12÷4=12 \div 4 =12÷4=
  5. −15÷5=-15 \div 5 =−15÷5=
  6. −18÷−6=-18 \div -6 =−18÷−6=
  7. 7×−2=7 \times -2 =7×−2=
  8. 21÷−7=21 \div -7 =21÷−7=
  9. −9×1=-9 \times 1 =−9×1=
  10. 0×−5=0 \times -5 =0×−5=

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  Medium Questions

  11. (−8)×6=(-8) \times 6 =(−8)×6=
  12. (−12)÷4=(-12) \div 4 =(−12)÷4=
  13. 5×(−5)×2=5 \times (-5) \times 2 =5×(−5)×2=
  14. (−30)÷(−5)×2=(-30) \div (-5) \times 2 =(−30)÷(−5)×2=
  15. (−4)×(−3)×(−2)=(-4) \times (-3) \times (-2) =(−4)×(−3)×(−2)=
  16. 24÷(−3)×(−2)=24 \div (-3) \times (-2) =24÷(−3)×(−2)=
  17. (−7)×(−6)÷2=(-7) \times (-6) \div 2 =(−7)×(−6)÷2=
  18. 9×(−2)×(−5)=9 \times (-2) \times (-5) =9×(−2)×(−5)=
  19. (−100)÷4÷(−5)=(-100) \div 4 \div (-5) =(−100)÷4÷(−5)=
  20. (−36)÷(−3)÷(−2)=(-36) \div (-3) \div (-2) =(−36)÷(−3)÷(−2)=

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  Hard Questions

  21. (−5)×4×(−2)×(−3)=(-5) \times 4 \times (-2) \times (-3) =(−5)×4×(−2)×(−3)=
  22. 144÷(−12)×(−3)=144 \div (-12) \times (-3) =144÷(−12)×(−3)=
  23. (−81)÷9×(−2)×(−5)=(-81) \div 9 \times (-2) \times (-5) =(−81)÷9×(−2)×(−5)=
  24. (−50)÷(−5)×(−4)×(−2)=(-50) \div (-5) \times (-4) \times (-2) =(−50)÷(−5)×(−4)×(−2)=
  25. (−8)×(−7)×(−6)×(−5)=(-8) \times (-7) \times (-6) \times (-5) =(−8)×(−7)×(−6)×(−5)=
  26. (−125)÷5×(−3)÷(−5)=(-125) \div 5 \times (-3) \div (-5) =(−125)÷5×(−3)÷(−5)=
  27. 64÷(−8)×(−4)×(−2)=64 \div (-8) \times (-4) \times (-2) =64÷(−8)×(−4)×(−2)=
  28. (−240)÷(−12)×(−5)÷2=(-240) \div (-12) \times (-5) \div 2 =(−240)÷(−12)×(−5)÷2=
  29. (−1000)÷(−10)×(−2)×(−1)=(-1000) \div (-10) \times (-2) \times (-1) =(−1000)÷(−10)×(−2)×(−1)=
  30. (−500)÷(−5)÷(−5)×(−2)=(-500) \div (-5) \div (-5) \times (-2) =(−500)÷(−5)÷(−5)×(−2)=

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  Super Hard Integer Problems

  31. (−9)×(−8)×(−7)×(−6)÷(−3)=(-9) \times (-8) \times (-7) \times (-6) \div (-3) =(−9)×(−8)×(−7)×(−6)÷(−3)=
  32. (−144)÷(−12)×(−11)÷(−3)=(-144) \div (-12) \times (-11) \div (-3) =(−144)÷(−12)×(−11)÷(−3)=
  33. (−5)×(−4)×(−3)×(−2)×(−1)=(-5) \times (-4) \times (-3) \times (-2) \times (-1) =(−5)×(−4)×(−3)×(−2)×(−1)=
  34. (−250)÷5×(−4)÷(−2)×(−3)=(-250) \div 5 \times (-4) \div (-2) \times (-3) =(−250)÷5×(−4)÷(−2)×(−3)=
  35. 1024÷(−4)×(−8)÷(−2)×(−1)=1024 \div (-4) \times (-8) \div (-2) \times (-1) =1024÷(−4)×(−8)÷(−2)×(−1)=
  36. (−999)×(−10)×(−2)÷(−5)=(-999) \times (-10) \times (-2) \div (-5) =(−999)×(−10)×(−2)÷(−5)=
  37. (−256)÷(−8)×(−4)×(−2)÷(−16)=(-256) \div (-8) \times (-4) \times (-2) \div (-16) =(−256)÷(−8)×(−4)×(−2)÷(−16)=
  38. (−2000)÷(−10)÷(−5)×(−4)×(−2)=(-2000) \div (-10) \div (-5) \times (-4) \times (-2) =(−2000)÷(−10)÷(−5)×(−4)×(−2)=
  39. (−10000)÷(−10)×(−10)×(−1)=(-10000) \div (-10) \times (-10) \times (-1) =(−10000)÷(−10)×(−10)×(−1)=
  40. (−150)×(−3)÷(−9)×(−8)÷(−2)=(-150) \times (-3) \div (-9) \times (-8) \div (-2) =(−150)×(−3)÷(−9)×(−8)÷(−2)=

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  Extreme Integer Challenges

  41. (−7)×(−6)×(−5)×(−4)×(−3)×(−2)×(−1)(-7) \times (-6) \times (-5) \times (-4) \times (-3) \times (-2) \times (-1)(−7)×(−6)×(−5)×(−4)×(−3)×(−2)×(−1)
  42. (−5000)÷(−50)×(−40)÷(−4)×(−2)(-5000) \div (-50) \times (-40) \div (-4) \times (-2)(−5000)÷(−50)×(−40)÷(−4)×(−2)
  43. (−12)×(−11)×(−10)×(−9)×(−8)÷(−6)(-12) \times (-11) \times (-10) \times (-9) \times (-8) \div (-6)(−12)×(−11)×(−10)×(−9)×(−8)÷(−6)
  44. (−1024)÷(−2)÷(−4)×(−8)×(−16)(-1024) \div (-2) \div (-4) \times (-8) \times (-16)(−1024)÷(−2)÷(−4)×(−8)×(−16)
  45. (−1)×(−2)×(−3)×(−4)×(−5)×(−6)×(−7)(-1) \times (-2) \times (-3) \times (-4) \times (-5) \times (-6) \times (-7)(−1)×(−2)×(−3)×(−4)×(−5)×(−6)×(−7)
  46. (−777)×(−7)×(−77)÷(−11)×(−2)(-777) \times (-7) \times (-77) \div (-11) \times (-2)(−777)×(−7)×(−77)÷(−11)×(−2)
  47. (−9999)÷(−9)×(−8)×(−7)×(−6)÷(−3)(-9999) \div (-9) \times (-8) \times (-7) \times (-6) \div (-3)(−9999)÷(−9)×(−8)×(−7)×(−6)÷(−3)
  48. (−20)×(−19)×(−18)×(−17)÷(−4)(-20) \times (-19) \times (-18) \times (-17) \div (-4)(−20)×(−19)×(−18)×(−17)÷(−4)
  49. (−100000)÷(−1000)×(−100)÷(−10)(-100000) \div (-1000) \times (-100) \div (-10)(−100000)÷(−1000)×(−100)÷(−10)
  50. (−555)×(−5)×(−55)×(−5555)÷(−5)(-555) \times (-5) \times (-55) \times (-5555) \div (-5)(−555)×(−5)×(−55)×(−5555)÷(−5)