Properties, Special Quantities and Operations

Factors

A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. All numbers have a factor of one since one multiplied by any number equals that number. All numbers can be divided by themselves to produce the number one. Therefore, we normally ignore one and the number itself as useful factors.

The number fifteen can be divided into two factors which are three and five.

The number twelve could be divided into two factors which are 6 and 2. Six could be divided into two further factors of 2 and 3. Therefore the factors of twelve are 2, 2, and 3.

If twelve was first divided into the factors 3 and 4, the four could be divided into factors of 2 and 2. Therefore the factors of twelve are still 2, 2, and 3.

* There are several clues to help determine factors. Any even number has a factor of two
* Any number ending in 5 has a factor of five
* Any number above 0 that ends with 0 (such as 10, 30, 1200) has factors of two and five.

To determine factors see if one of the above rules apply (ends in 5, 0 or an even number). If none of the rules apply, there still may be factors of 3 or 7 or some other number.

Prime and Composite Numbers

A prime number is a whole number that only has two factors which are itself and one. A composite number has factors in addition to one and itself.

The numbers 0 and 1 are neither prime nor composite.

All even numbers are divisible by two and so all even numbers greater than two are composite numbers.

All numbers that end in five are divisible by five. Therefore all numbers that end with five and are greater than five are composite numbers.

The prime numbers between 2 and 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number that is a common factor of two or more numbers.

How to find the greatest common factor:

* Determine if there is a common factor of the numbers. A common factor is a number that will divide into both numbers evenly. Two is a common factor of 4 and 14.
* Divide all of the numbers by this common factor.
* Repeat this process with the resulting numbers until there are no more common factors.
* Multiply all of the common factors together to find the Greatest Common Factor


Simplifying Fractions

Fractions may have numerators and denominators that are composite numbers (numbers that has more factors than 1 and itself).

How to simplify a fraction:

* Find a common factor of the numerator and denominator. A common factor is a number that will divide into both numbers evenly. Two is a common factor of 4 and 14.
* Divide both the numerator and denominator by the common factor.
* Repeat this process until there are no more common factors.
* The fraction is simplified when no more common factors exist.

Another method to simplify a fraction

* Find the Greatest Common Factor (GCF) of the numerator and denominator
* Divide the numerator and the denominator by the GCF

Least Common Multiple

The Least Common Multiple (LCM) is the smallest number that two or more numbers will divide into evenly.

How to find the Least Common Multiple of two numbers:

* Find the Greatest Common Factor (GCF) of the numbers
* Multiply the numbers together
* Divide the product of the numbers by the GCF.

Example: Find the LCM of 15 and 12

* Determine the Greatest Common Factor of 15 and 12 which is 3
* Either multiply the numbers and divide by the GCF (15*12=180, 180/3=60)
* OR - Divide one of the numbers by the GCF and multiply the answer times the other number (15/3=5, 5*12=60)

Least Common Denominator

The Least Common Denominator (LCD) is the Least Common Multiple of two or more denominators.

How to find the Least Common Denominator:

* Find the Greatest Common Factor of the denominators.
* Multiply the denominators together.
* Divide the product of the denominators by the Greatest Common Factor.

Example: Find the LCD of 2/9 and 3/12

* Determine the Greatest Common Factor of 9 and 12 which is 3
* Either multiply the denominators and divide by the GCF (9*12=108, 108/3=36)
* OR - Divide one of the denominators by the GCF and multiply the quotient times the other denominator (9/3=3, 3*12=36)

How to rename fractions and use the Least Common Denominator:

* Divide the LCD by one denominator.
* Multiply the numerator times this quotient.
* Repeat the process for the other fraction(s)

Example: Add 2/9 + 3/12

* LCD is 36
* First fraction (2/9): 36/9 = 4, 4*2 = 8, first fraction is renamed as 8/36
* Second fraction (3/12): 36/12 = 3, 3*3 = 9, second fraction is renamed as 9/36
* It is possible to add or subtract fractions that have the same denominator
* 8/36 + 9/36 = 17/36

Estimating by Front End Estimation

Front end estimation mostly produces a closer estimate of sums or differences than the answer produced by adding or subtracting rounded numbers.

How to estimate a sum by front end estimation:

* Add the digits of the two highest place values
* Insert zeros for the other place values


* Example 1: 4496 + 3745 is estimated to be 8100 by front end estimation (i.e. 4400 + 3700).
* Example 2: 4496 + 745 is estimated to be 5100 by front end estimation (i.e. 4400 + 700).


Estimating by front end estimation

Front end estimation generally produces a better estimate of sums or differences than rounding before adding or subtracting.

How to estimate a difference by front end estimation:

* Subtract the digits of the two highest place values
* Insert zeros for the other place values
*
* Example: 7396 minus 3745 is estimated to be 3600 by front end estimation (i.e. 7300-3700).


Calculations using the Order of Operations

The order of operations are rules that govern which mathematical operations are done first.

* Do operations in parentheses and other grouping symbols first. If there are grouping symbols within other grouping symbols do the innermost first.
* Do multiplication and division operations from left to right.
* Do addition and subtraction operations from left to right

Example: 2+3*(4+(6*3-8))*2
2+3*(4+(18-8))*2
2+3*(4+10)*2
2+3*14*2
2+42*2
2+84
86


Properties of Addition

There are four mathematical properties which involve addition. The properties are the commutative, associative, additive identity and distributive properties.

Commutative property: When two numbers are added, the sum is the same regardless of the order of the addends. For example 4 + 2 = 2 + 4

Associative Property: When three or more numbers are added, the sum is the same regardless of the grouping of the addends. For example (2 + 3) + 4 = 2 + (3 + 4)

Additive Identity Property: The sum of any number and zero is the original number. For example 5 + 0 = 5.

Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3


Multiplication Properties

There are four properties involving multiplication that will help make problems easier to solve. They are the commutative, associative, multiplicative identity and distributive properties.

Commutative property: When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands. For example 4 * 2 = 2 * 4

Associative Property: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example (2 * 3) * 4 = 2 * (3 * 4)

Multiplicative Identity Property: The product of any number and one is that number. For example 5 * 1 = 5.

Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3