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Lesson 1: Properties of numbers

# Properties of multiplication

Explore the commutative, associative, and identity properties of multiplication.
In this article, we'll learn the three main properties of multiplication. Here's a quick summary of these properties:
Commutative property of multiplication: Changing the order of factors does not change the product. For example, 4, times, 3, equals, 3, times, 4.
Associative property of multiplication: Changing the grouping of factors does not change the product. For example, left parenthesis, 2, times, 3, right parenthesis, times, 4, equals, 2, times, left parenthesis, 3, times, 4, right parenthesis.
Identity property of multiplication: The product of 1 and any number is that number. For example, 7, times, 1, equals, 7.

## Commutative property of multiplication

The commutative property of multiplication says that changing the order of factors does not change the product. Here's an example:
4, times, 3, equals, 3, times, 4
Notice how both products are 12 even though the ordering is reversed.
Here's another example with more factors:
1, times, 2, times, 3, times, 4, equals, 4, times, 3, times, 2, times, 1
Notice that both products are 24.
Which of these is an example of the commutative property of multiplication?

## Associative property of multiplication

The associative property of multiplication says that changing the grouping of the factors does not change the product. Here's an example:
start color #11accd, left parenthesis, 2, times, 3, right parenthesis, times, 4, end color #11accd, equals, start color #e07d10, 2, times, left parenthesis, 3, times, 4, right parenthesis, end color #e07d10
Remember that parentheses tell us to do something first. So here's how we evaluate the left-hand side:
empty space, start color #11accd, left parenthesis, 2, times, 3, right parenthesis, times, 4, end color #11accd
equals, 6, times, 4
equals, 24
And here's how we evaluate the right-hand side:
empty space, start color #e07d10, 2, times, left parenthesis, 3, times, 4, right parenthesis, end color #e07d10
equals, 2, times, 12
equals, 24
Notice that both sides equal 24 even though we multiplied the 2 and the 3 first on the left-hand side, and we multiplied the 3 and the 4 first on the right-hand side.
Which of these is an example of the associative property of multiplication?