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Multiplying & Dividing Integers

Multiplying and dividing integers is easier than adding them, because there is only one thing to decide: the sign of the answer. Same signs give a positive answer; different signs give a negative answer. The digits themselves multiply or divide exactly the way they always have.

So −6⋅(−7)-6 \cdot (-7) is just 6⋅7=426 \cdot 7 = 42 with a sign decision stapled on: both factors are negative, signs match, answer is positive 4242. Two small checks — multiply the absolute values, then set the sign — and you're done.

The sign rules

Same signs, positive answer: 5⋅4=205 \cdot 4 = 20 and −5⋅(−4)=20-5 \cdot (-4) = 20. Different signs, negative answer: −5⋅4=−20-5 \cdot 4 = -20 and 5⋅(−4)=−205 \cdot (-4) = -20.

Division follows the identical rules, because division is just multiplication asked backwards. −36−9=4\dfrac{-36}{-9} = 4 (same signs, positive) and 42−6=−7\dfrac{42}{-6} = -7 (different signs, negative).

Why a negative times a negative is positive

Think of multiplying by −1-1 as flipping a number to the opposite side of zero. One flip: −1⋅7=−7-1 \cdot 7 = -7. Flip it again: −1⋅(−7)=7-1 \cdot (-7) = 7 — two flips land you back on the positive side. Every negative factor is one flip, so two negative factors flip twice and come back positive.

That flip idea also handles longer products. Count the negative factors: an even count means the flips cancel out and the answer is positive; an odd count leaves one flip standing and the answer is negative. (−2)(−3)(−4)(-2)(-3)(-4) has three negative factors, so it is negative: −24-24.

Keep the rules straight: multiplying is not adding

The most common mistake is dragging the addition rules into a multiplication problem. Two negatives added stay negative (−6+(−7)=−13-6 + (-7) = -13), but two negatives multiplied turn positive (−6⋅(−7)=42-6 \cdot (-7) = 42). Before you set the sign, look at the operation.

Worked examples

Example 1: different signs

Multiply −4⋅7-4 \cdot 7.

Multiply the absolute values4⋅7=284 \cdot 7 = 28
Different signs: the answer is negative−4⋅7=−28-4 \cdot 7 = -28

Answer: −28-28

Example 2: same signs

Multiply −6⋅(−5)-6 \cdot (-5).

Multiply the absolute values6⋅5=306 \cdot 5 = 30
Same signs: the answer is positive−6⋅(−5)=30-6 \cdot (-5) = 30

Answer: 3030

Example 3: dividing with matching signs

Divide −36−9\dfrac{-36}{-9}.

Divide the absolute values36÷9=436 \div 9 = 4
Same signs: the answer is positive−36−9=4\dfrac{-36}{-9} = 4

Answer: 44

Example 4: dividing with different signs

Divide 42−6\dfrac{42}{-6}.

Divide the absolute values42÷6=742 \div 6 = 7
Different signs: the answer is negative42−6=−7\dfrac{42}{-6} = -7

Answer: −7-7

Try one yourself

Common questions

Are the sign rules the same for multiplication and division?

Yes, exactly the same. Same signs give a positive answer, different signs give a negative answer — whether you multiply or divide.

What if there are three or more factors?

Count the negative factors. An even count gives a positive product; an odd count gives a negative product. For example, (−2)(−3)(−4)=−24(-2)(-3)(-4) = -24 because three negatives is an odd count.

Why is −6+(−7)-6 + (-7) negative but −6⋅(−7)-6 \cdot (-7) positive?

Because addition and multiplication follow different rules. Adding two negatives stacks the debt: −13-13. Multiplying by a negative flips the sign, and two flips land positive: 4242. Always check the operation before setting the sign.

Does the order of the signs matter, like −5⋅4-5 \cdot 4 versus 5⋅(−4)5 \cdot (-4)?

No. Both have one negative factor and one positive factor, so both are negative: −20-20. Only the mix of signs matters, not which position they sit in.

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