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What Is a Product in Math?
A product is the result of multiplying two or more numbers together. In 7 × 6 = 42, the product is 42 and the numbers multiplied are called factors. This guide covers the definition, how product differs from sum, difference and quotient, worked examples with fractions and negative numbers, and the way products and factors are the same statement read in opposite directions. The factor calculator runs that relationship backwards for any number.
6 min read
What does product mean in math?
A product is the result of multiplying two or more numbers together. In 7 × 6 = 42, the product is 42. The numbers being multiplied are called factors, so 7 and 6 are factors of 42.
That is the entire definition, and it is worth stating plainly because the word is almost always met in a sentence like “find the product of 7 and 6”, where the instruction is simply multiply them.
Product vs sum vs difference vs quotient
Each of the four basic operations has its own word for its result, and mixing them up is the most common reason a correct calculation earns no marks. They are worth learning as a set:
| Word | Operation | Example |
|---|---|---|
| Product | Multiplication | 7 × 6 = 42 |
| Sum | Addition | 7 + 6 = 13 |
| Difference | Subtraction | 7 − 6 = 1 |
| Quotient | Division | 42 / 6 = 7 |
“The product of 7 and 6” is 42. “The sum of 7 and 6” is 13. A question that asks for one and receives the other is wrong even though the arithmetic was performed correctly, which is why the vocabulary is worth five minutes.
Worked examples
The word covers more than whole numbers, so it is worth seeing it in three forms.
- Whole numbers. The product of 7 and 6 is 7 × 6 = 42.
- Fractions. Multiply the numerators, then the denominators: 2/3 × 3/4 = 6/12, which simplifies to 1/2. Note the product is smaller than either factor, which surprises people who expect multiplication to make things bigger.
- Negatives. −5 × 3 = −15, but −5 × −3 = 15. Two negative factors give a positive product.
The fraction case is the one that trips people. Multiplying by any value between 0 and 1 makes the result smaller, so “multiplication makes things bigger” is a rule that only holds for factors above 1.
Products and factors are one statement
Every product statement can be read backwards as a factor statement. 7 × 6 = 42 says that 42 is the product of 7 and 6, and equally that 7 and 6 are factors of 42. Nothing changes but the direction you read it.
This is why finding factors and finding products are taught together. Multiplying is the easy direction; going backwards from 42 to all the pairs that produce it — 1 and 42, 2 and 21, 3 and 14, 6 and 7 — is the harder one, and it is what the factor calculator does.
A prime number is one whose only factor pair is 1 and itself, which is the same as saying it can be written as a product in only one way.
Products worth recognising
- Anything times zero is zero. The zero product property runs the other way too: if a product equals 0, at least one factor must be 0. That is the step that lets you solve (x − 2)(x + 5) = 0 by setting each bracket to zero.
- Anything times one is itself. One is the multiplicative identity.
- Product of powers. Multiplying powers of the same base adds the exponents: 2³ × 2⁴ = 2⁷ = 128.
- Order does not matter. 7 × 6 and 6 × 7 give the same product. Multiplication is commutative, unlike subtraction and division.
Common mistakes
- Adding when asked for a product. The single most common slip, and the reason this vocabulary matters.
- Assuming a product is larger than its factors. True for factors above 1, false for fractions and for zero.
- Losing a sign. An odd number of negative factors gives a negative product; an even number gives a positive one.
- Multiplying exponents instead of adding them when the bases match.
Quick summary
- A product is the result of multiplying; the numbers multiplied are factors.
- Product, sum, difference and quotient name the results of the four operations.
- A product can be smaller than its factors when a factor is between 0 and 1.
- If a product is zero, at least one factor is zero.
The factor calculator runs the relationship backwards, listing every factor pair whose product is the number you enter.