Practice Maths

Multiplying & Dividing Integers

Key Terms

same signs
When both numbers are positive or both negative — the product or quotient is positive: (−4) × (−3) = +12.
different signs
When one number is positive and the other negative — the product or quotient is negative: 4 × (−3) = −12.
even number of negatives
A product with an even count of negative factors gives a positive result.
odd number of negatives
A product with an odd count of negative factors gives a negative result.

Sign rules for × and ÷: same signs = positive  |  different signs = negative  |  same rules apply for division.

Powers of negatives: (−2)2 = (−2) × (−2) = +4, but −22 = −(2 × 2) = −4. The brackets matter!

Worked Examples

Example 1: (−6) × (−4) = +24   (negative × negative = positive)

Example 2: (−35) ÷ 7 = −5   (negative ÷ positive = negative)

Example 3: (−3)2 = (−3) × (−3) = 9, but −32 = −9

Why the Sign Rules Work

The sign rules for multiplication might seem arbitrary, but they're not — they follow logically from patterns. Consider 3 × 4 = 12, 2 × 4 = 8, 1 × 4 = 4, 0 × 4 = 0. Each step decreases by 4. Continuing: (−1) × 4 = −4, (−2) × 4 = −8, (−3) × 4 = −12. The pattern forces positive × negative = negative. Now take (−3) × 4 = −12, (−3) × 3 = −9, (−3) × 2 = −6, (−3) × 1 = −3, (−3) × 0 = 0. Continuing: (−3) × (−1) = +3, (−3) × (−2) = +6. The pattern forces negative × negative = positive. These rules aren't arbitrary — they're forced by consistency of arithmetic.

The Four Sign Combinations

Positive × positive = positive: 5 × 6 = 30. (Both "forward" movements; result moves forward.) Positive × negative = negative: 5 × (−6) = −30. Negative × positive = negative: (−5) × 6 = −30. Negative × negative = positive: (−5) × (−6) = 30. The shortcut: if the signs are the same (both + or both −), the result is positive. If the signs are different, the result is negative. This works for division too: (−30) ÷ 6 = −5; (−30) ÷ (−6) = 5.

Key Tip: Count the number of negative signs in a multiplication. An even number of negatives gives a positive result; an odd number gives a negative result. So (−2) × (−3) × (−4): three negatives (odd), result is negative: −24.

Powers of Negative Numbers

This is where brackets matter critically. (−3)2 = (−3) × (−3) = +9. But −32 = −(3 × 3) = −9. The difference: in (−3)2, the negative is part of what's being squared. In −32, only 3 is being squared, then negated. The bracket changes the meaning entirely.

Odd powers of negatives are negative: (−2)3 = (−2) × (−2) × (−2) = 4 × (−2) = −8 (three negatives = odd = negative). Even powers of negatives are positive: (−2)4 = 16 (four negatives = even = positive). This is why even powers of negative numbers always give positive results.

Key Tip: (−3)2 = 9 (positive) but −32 = −9 (negative). The brackets make all the difference. In (−3)2, the negative is inside the power — it gets squared. In −32, the negative is outside — it's applied after squaring. Always check for brackets before evaluating.

Real-World Applications

Multiplication of integers appears in practical calculations. Temperature: a temperature drops 3°C per hour for 5 hours: change = −3 × 5 = −15°C. Debt: you owe $45 per week for 6 weeks: total debt = −45 × 6 = −$270. Division: if a temperature changed −24°C over 8 hours, the rate is −24 ÷ 8 = −3°C per hour. In physics, displacement and velocity calculations use signed quantities — positive and negative integers give direction as well as magnitude.

Mastery Practice

  1. Multiply Integers Fluency

    1. (−3) × 7
    2. (−5) × (−4)
    3. 8 × (−6)
    4. (−9) × (−3)
    5. (−2) × 11
    6. (−7) × (−7)
    7. 4 × (−12)
    8. (−10) × 5

  2. Divide Integers Fluency

    1. (−24) ÷ 6
    2. (−36) ÷ (−9)
    3. 40 ÷ (−8)
    4. (−55) ÷ 5
    5. (−48) ÷ (−6)
    6. 72 ÷ (−9)
    7. (−63) ÷ 7
    8. (−100) ÷ (−10)

  3. Powers of Negative Numbers Fluency

    1. (−3)2
    2. −32
    3. (−2)3
    4. −42
    5. (−5)2
    6. (−1)5
    7. (−2)4
    8. −24

  4. Determine the Sign First Understanding

    Without calculating, state whether the answer is positive or negative:

    1. (−3) × (−5) × 2
    2. (−4) × (−6) × (−1)
    3. (−7)2
    4. −72
    5. (−2) × 8 × (−3)
    6. (−10) ÷ (−2) × (−1)

  5. Find the Missing Integer Understanding

    Find the value of n:

    1. n × (−4) = −20
    2. (−6) × n = 42
    3. n ÷ (−3) = 8
    4. (−36) ÷ n = −9
    5. n × (−7) = 0
    6. (−5) × n = −45
    7. n ÷ 4 = −12
    8. 54 ÷ n = −6

  6. Real-World Problem Solving Problem Solving

    1. A business records a loss of $150 each day for 8 days. Use integers to express the total change in the business account.
    2. The temperature inside a freezer drops 3°C per hour. After 6 hours, what is the total temperature change?
    3. A share loses $4 in value each week. After how many weeks will it have lost a total of $52 in value?
    4. A submarine descends at a rate of 15 m per minute. Starting at the surface (0 m), what depth does it reach after 7 minutes? Express your answer as an integer.
    5. Sophie owes equal amounts to 5 friends. Her total debt is −$85. How much does she owe each friend?
    6. A scientist records the temperature change in a chemical reaction as −2°C every 30 seconds. What is the total temperature change after 8 minutes?

  7. Mixed Multiplication and Division Fluency

    1. (−4) × 3 ÷ (−6)
    2. (−48) ÷ (−4) × (−2)
    3. 30 ÷ (−5) × (−3)
    4. (−7) × (−3) ÷ 7
    5. (−2) × 6 ÷ (−4)
    6. (−9) × (−4) ÷ (−6)
    7. (−60) ÷ 12 × (−5)
    8. 5 × (−8) ÷ (−10)

  8. Explain the Sign Rules Understanding

    1. A student says (−3) × (−4) = −12 because “there are two negatives so the answer is negative”. Identify the error and give the correct answer.
    2. Explain why (−1)6 is positive but (−1)7 is negative.
    3. Without calculating, predict the sign of (−2) × (−3) × (−1) × 4. Then calculate to check.
    4. Explain why dividing a negative by a negative always gives a positive result, using a real-world example.

  9. Integer Patterns and Sequences Understanding

    Each sequence below is geometric (multiply by a constant). Find the next three terms and state the multiplier:

    1. 2, −6, 18, −54, …
    2. −1, 3, −9, 27, …
    3. 64, −32, 16, −8, …
    4. −5, 10, −20, 40, …

  10. Extended Real-World Problems Problem Solving

    1. A mining company removes ore at a rate of 250 tonnes per day. Using a negative integer to represent removal, write an expression for the total change in ore after 12 days, and calculate it.
    2. A scientist measures the temperature in a storage facility. The temperature drops by 4°C every hour for the first 6 hours and then rises by 2°C every hour for the next 3 hours. What is the net change in temperature?
    3. A share price changes by −$3 each day for 5 days, then by +$4 each day for 4 days. If the share started at $50, what is its final price?
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