For any numbers a and b, a + b = b + a

Commutative in French means "switchable", therefore we can "switch" the numbers in addition. This may also be known as the order property, where the order of the numbers are different. For example:

35 + 21 = 21 + 35
56 = 56

For any numbers a and b, ab = ba

For example:

5*12 = 12*5
60 = 60

For any numbers a, b, and c, (a + b) + c = a + (b + c)

This may also be known as the grouping property. Notice that the numbers are grouped usually using parenthesis, ( ), or brackets, [ ]. Also, the order of the numbers need to stay the same. For example:

(2 + 4) + 6 = 2 + (4 + 6)
6 + 6 = 2 + 10
12 = 12

For any numbers a, b, and c, (AB)c = a(bc) = abc

For example:

(2*3) * 4 = 2* (3*4) = 2*3*4
6*4 = 2*12 = 24
24 = 24 = 24

Because multiplication is both commutative and associative, as we can multiply the numbers in any order without affecting the product, it can help shorten multiplications. Notice that the numbers are grouped usually using parenthesis, ( ), or brackets, [ ]. Also, the order of the numbers need to stay the same. For example:

Multiply: 9 * 25 * 4 * 2 * 25 * 4

Because of the associative property we can group this problem so it can be completed easily in our head.

9 * 2 * (4 * 25) * (4 * 25)
18 * 100 * 100
18 * 10000
180000

For any number a, a + 0 = a

For example:

44 + 0 = 44

For any number a, a * 1 = a

For example:

35 * 1 = 35

For any number a, a * 0 = 0

For example:

75 * 0 = 0

For any number a, b, and c, c(a + b) = c*a + c*b

This works over addition like above and for subtraction as well. c(a - b) = c*a - c*b

For example:

4(2 + 3) = 4*2 + 4*3
4(5) = 8 + 12
20 = 20

A composite number is any number exactly divisible by one or more numbers other than itself and 1. A composite number is a positive integer that is not prime nor equal to 1.

For example:

35 is a composite number because it is divisible by 5 and 7 as well as being divisible by 1 and itself, 35.

A prime number is a positive integer other than 1 that is divisible only by itself and 1.

For example:

13 is a prime number because it is only divisible by 1 and itself. There are no other factors for 13 except 1 and 13. 3, 7, 11, 17, 19, and 23 including 13 are the first few prime of numbers.

Finding the LCM or least common denominator (LCD) helps with finding common denominators for adding or subtracting fractions. There are a couple of different ways to find the LCM.

One way is to list the multiples of the numbers you are working with and then find the first multiple that is common to those numbers. For example to find the LCM of 8, 20, and 40 we would list the multiples for each.

• 8 - 8, 16, 24, 32, 40
• 20 - 20, 40, 60, 80
• 40 - 40, 80

  The least common multiple of 8, 20, and 40 is 40.

Another way to find the LCM is to factor each number into prime factors.

• 8 - 2 x 2 x 2
• 20 - 2 x 2 x 5
• 40 - 2 x 2 x 2 x 5

List once, each prime factor that appears in the factorizations.

2, 5

Raise each prime factor to its highest power in the factorization.

• The factor 2 appears 3 times as the highest in 40.
• The factor 5 appears once in each of two factorizations.
• 2 x 2 x 2, and 5

Multiply these factors to produce the LCM.

2 x 2 x 2 x 5 = 40

We arrive at the same answer, 40.

To find the greatest common factor (GCF) of a set of numbers is to list the factor pairs for each. For example to find the GCF of 21 and 36 we would list the factors of each.

• 21 - 1 * 21, 3 * 7
• 36 - 1 * 36, 2 * 18, 3 * 12, 4 * 9

Let's put the factors in order from least to greatest.

• 21 - 1, 3, 7, 21
• 36 - 1, 2, 3, 4, 9, 12, 18, 36

Now look for the largest, greatest, factor that is common to both. In this case, 3 is the greatest common factor.

• A number is divisible by 2 if the digit in the ones place is 0, 2, 4, 6, or 8.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if the digit in the ones place is 0 or 5.
• A number is divisible by 9 if the sum of its digits is divisible by 9.
• A number is divisible by 10 if the digit in the ones place is 0.