The product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.
An array shows a product as equal groups: 4 rows of 6 make 24. The numbers you multiply are factors; the answer is the product.
Formal definition and notation
For numbersa and b, the product a × b (also written a · b, (a)(b) or ab) is the result of multiplying them. For whole numbers, a × b is the total in a equal groups of b. A product can have any number of factors, such as 2 × 3 × 5 = 30, and in algebra a product can contain variables, as in 7xy.
Symbol
Read it as
Meaning
a × b
a times b
the usual sign in arithmetic
a · b
a times b
the raised dot, used once letters appear, because × looks like x
(a)(b) or a(b)
a times b
parentheses next to each other mean multiply
ab or 3x
a b, three x
a number or letter written right next to a letter means multiply
Key facts
Times 0: any number times 0 has a product of 0, so 845 × 0 = 0.
Times 1: any number times 1 has itself as the product, so 17 × 1 = 17.
Signs: the product of two integers with the same sign is positive, and with different signs is negative: (−3)(−4) = 12, but (−3)(4) = −12.
Multiplying by a number between 0 and 1 makes the product smaller than the other factor: 10 × 0.5 = 5.
The opposite operation is division: 24 ÷ 6 = 4 undoes 4 × 6 = 24. The answer to a division is the quotient.
How to find a product
How to find a product
factor × factor = product
factor
each number being multiplied
product
the result of the multiplication
Order does not matter: 4 × 6 and 6 × 4 have the same product, 24. Group factors any way you like: (2 × 5) × 7 = 2 × (5 × 7) = 70.
Try it: build a product as an array
Interactive
Change the number of rows and columns. The number of unit squares inside is the product of the two side lengths.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Product examples, step by step
Example 1: Product of two numbers
Warm-up
Find the product of 8 and 9.
“Product” means multiply: 8 × 9.
Use a known fact: 8 × 10 = 80.
Take away one 8: 80 − 8 = 72.
Answer: The product of 8 and 9 is 72.
Example 2: Product of three factors with a negative
Core
Find the product (−5)(4)(−3).
Multiply the first two factors: (−5)(4) = −20. Different signs give a negative.
Multiply by the last factor: (−20)(−3) = 60. Same signs give a positive.
Check with a sign count: two negative factors (an even number) make the product positive.
Answer: The product is 60.
Example 3: Product in a word problem
Test-style
The product of a number and 6 is 7 more than 5 times the number. What is the number?
Let the number be n. “The product of a number and 6” is 6n.
Product always means multiply. The answer 8 is the sum of 5 and 3, which comes from adding. Tests use “sum” and “product” side by side to catch this.
✗ Mistake7 × 0 = 7, because multiplying by 0 “does nothing”.
✓ Instead7 × 0 = 0.
Seven groups of nothing is nothing. Adding 0 leaves a number unchanged; multiplying by 0 always gives 0. Multiplying by 1 is what leaves a number unchanged.
✗ MistakeThe product of −4 and −6 is −24.
✓ Instead(−4)(−6) = 24.
Two negative factors give a positive product. Count the negative signs: an even count makes the product positive, an odd count makes it negative.
✗ MistakeReading 3(4) as 34 or as 3 + 4.
✓ Instead3(4) = 3 × 4 = 12.
A number written right next to parentheses means multiply. The same rule makes 3x mean 3 times x.
A quotient is the answer to a division, the opposite of a product. 24 ÷ 6 = 4 has quotient 4, and 4 × 6 = 24 has product 24.
Quick check: 3 questions
Question 1 of 3
What is the product of 12 and 5?
Show answer and explanation
Answer: C. Product means multiply: 12 × 5 = 60. 17 is the sum, 7 is the difference and 2.4 is the quotient 12 ÷ 5.
Question 2 of 3
Which expression is the product of x and 9?
Show answer and explanation
Answer: A. A number written next to a letter means multiply, so 9x is 9 times x. The other choices add, subtract or divide.
Question 3 of 3
What is (−2)(−3)(−5)?
Show answer and explanation
Answer: C.(−2)(−3) = 6, then 6 × (−5) = −30. Three negative factors (an odd count) make the product negative. 30 forgets one sign; −10 and 10 add instead of multiply.
Product: frequently asked questions
Does product mean add or multiply?
Multiply. The product is always the result of multiplication, so “the product of 6 and 7” is 42. Adding gives a sum (13), subtracting gives a difference and dividing gives a quotient. On tests, look for the word “of” after product: “the product of” names the factors to multiply, as in “the product of a number and 4”, which is 4n.
Can a product be smaller than the numbers you multiply?
Yes. If one factor is between 0 and 1, the product is smaller than the other factor: 8 × 12 = 4. If a factor is negative, the product can be less than both factors: 3 × (−2) = −6. Multiplying only makes things bigger when both factors are greater than 1.
What is the product of a number and 0?
The product of any number and 0 is 0. Think of 9 groups with nothing in each group: there is nothing in total. This is called the multiplication property of zero, and it is why a product such as 25 × 14 × 0 × 3 is 0 without any work.
FactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. 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A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2SumA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2, Grade 3MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2