Download prime factorization

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FACTORS
A factor is a number or numbers that another number is divisible by. For instance every number
has at least two “Factors”, the number itself and the number 1.
A “Prime” number is any number that is only divisible by itself and the number 1. However a
“Prime” number can be “Factor” of another number. A simple example can be shown with the
number “7”. “7” itself is only divisible by “1” and “7” making it “Prime”, however the number
“21” contains four factors one of which is the number “7” along with “1”, “3” and, “21”.
Numbers that have more than two factors are known as composites
An easier way to find factors of a number is to draw it out on paper in “rainbow” form.
Take a number, 180 for example; Based on the rules above we already know that at one end of
the spectrum is the number 1 and the number at the other end has to be 180. This holds true not
only because all numbers have the factor of 1 and itself, but 180 multiplied by 1 equals 180. The
next step is to fins the next highest number that is divisible by 180. In this case it is 90, and 90
multiplied by 2 is 180. So, the number 90 as well as two are placed at the appropriate end of the
spectrum so that they are in ascending order leaving room in the middle for more factors.
So at this point we know to find all the factors of a number, we keep looking for the highest
divisible number in descending order of the number we are studying and its multiple that equals
the studied number. When the numbers meet in the middle of the spectrum and no other divisible
numbers can be found, we know that we have identified all the factors of the studied number.
Now, a few short cuts in the process. As we are identifying factors, in this case the number 180,
there are a few “magic” numbers in the factors. In factor numbers with multiple digits, if we add
the digits together and they equal 9 we know that the number 3 and 9 are factors.
For example the number 180, 1 + 8 + 0 = 9, so we know at this point 3 and 9 are factors.
The same would hold true for the number 72, 7 + 2 = 9
In the case of larger numbers we need to keep adding digits until we come to a one digit number.
For example the number 180,180 we would add 1 + 8 + 0 + 1 + 8 + 0 = 18, now we would add
the two digits 1 + 8 = 9. We now know that not only are 3 and 9 factors of 180,180 but the
number 18 is as well!
Now for an easy one. Any number ending in a 5 or a 0 always has the number 5 as a factor. Any
number ending in 0 has 10 as a factor.
The number 6 can always be added as a factor if the number 3 and 2 are factors.
These tricks with smaller numbers can be a significant help when coupled with their multiple to
identify larger factors.
PRIME FACTORIZATION
These numbers are as simple as the definition, factors of a number that in and of them selves are
prime. To help us here, rather than a rainbow, we will construct a “factor tree”.
In this case take the number 78,102:
As before we will find the largest factor, in this case it is 13,017 which when multiplied by 6
equals the studied number of 78,102.
78,102
6
and
13,017
neither of these numbers is prime, thus not part of the prime factorization, so we need to break
them down further: The factors of 6 are 2 and 3. 2 and 3 are both prime so they are part of the
prime factorization of 78,102. we are now done with this side of the “tree”.
78,102
6
and
13,017
2 and 3
Now for the other side: 13,017 is divisible by 3 and 4,339. We already know that 3 is prime and
is part of the prime factorization, so we shift our focus to the larger remaining number 4,339.
78,102
6
and
2 and 3
and
13,017
3 and 4,339
At this point we will find that the number 4,339 is only divisible by 1 and itself making it prime.
We need to always avoid being fooled that a large number can not be prime. In this case we have
a four digit prime number, making our tree look as follows (prime numbers are in BOLD):
78,102
6
and
2 and 3
and
13,017
3 and 4,339
So, our prime factorization for 78,102 would read as follows:
2, 3, 3, 4,339
EXPONENTIAL NOTATION
Here we have a fancy name for a relatively easy operation. In an equation that contains and
number shown like this:
42 (4 squared)
it is simply 4 x 4. So
42 is in fact also the number
16.
With exponential notation the smaller number (in this case 2) is called the “exponent”. To solve
a figure with an exponent we simply take the number (in this case 4) and multiply it by itself as
many times as the exponent indicates.
For example:
Z4
+
Y2 is equal to Z X Z X Z X Z
+
Y X Y and vice versa.
ESTIMATION
Estimation is a valuable tool in everyday life. Shopping for example, it would be both foolish
and time consuming. Rather than adding numbers like 59 cents, 1.27, 2.99, 1.18, etc and adding
applicable taxes it is much easier to estimate the total amount of money you may need.
Estimation is simple, the easiest way is to round to the nearest whole number. Using the figures
above 59 cents would round to 1.00, 1.27 also to 1.00, 2.99 to 3.00 and 1.18 to 1.00. This will
give one a better idea or estimation of how much money they can expect to spend.
It also can help with things such as attendance. If there were three separate groups of people
scheduled to attend a conference, say one of 489, one of 724, and one of 317 you would have a
total of 1530 expected people. The person in charge of catering would not order food for an
“exact” number of guests, rather would estimate the crowd. 489 would round to 500, 724 to 700,
and 317 to 300 ending with a grand total of 1500. This is a good “estimation” of attendance for
the sake of ordering food for what is essentially an unknown number of people that may or may
not be hungry.
PROPERTIES
There are several “properties” in modern mathematics. The first discussed here is the addition
property of 0, simply it means any number added to zero is equal to that number.
For example:
read:
a+0=a
if a was representative of the number 7 then the equation would
7+0=7
The same holds true for the multiplication property of 1, any number multiplied by 1 equals the
number.
For example:
ax1=a
equation would read :
if a was representative of the number 45 then the
45 x 1 = 45
There is also a multiplication property of 0, meaning simply that any number multiplied by zero
is equal to 0.
For example:
ax0=0
If a was representative of the number 100 the equation would read
100 x 0 = 0
In both addition and multiplication the commutative property exists, put simply moving the
numbers around within an equation would yield the same result.
For example:
a + b would have the same meaning as
In multiplication the same holds true:
b+a
a x b is equal to b x a.
(i.e. a + b is equal to b + a)
In the above equations if a was representative of the number 14 and b was representative of the
number 3 the equations would read as:
14 + 3 = 17
as
3 + 14 = 17
As well as 14 x 3 = 42 the same as 3 x 14 = 42.
As algebra comes more complex the “associative” property can be more relevant. In both
addition and multiplication the associative property allows the parenthesis or brackets to by
moved within the context of an equation.
For example (a + b) + c will yield the same result as a + (b + c) as (a x b) x c would be equal
to a x (b x c).
Using the following key: a = 2, b = 3, and c = 4
the above equations would translate to the following:
(2 + 3) + 4 is the same as 2 + (3 + 4) both equal to 9
and
(2 x 3) x 4 is the same as 2 x (3 x 4) both equal to 24
The final property we will discuss on this page is the “distributive” property, this is most simply
explained when a number outside parenthesis or brackets is distributed to the numbers inside the
parenthesis or brackets.
For example: in the equation a (b + c), a is distributed to by multiplied by both b and c and the
two numbers are then added together.
For example: a (b + c) would translate into a x b + a x c
Using the following key: a = 2, b = 3, and c = 4
the equations would read as:
2 (3 + 4) translating also to 2 x 3 + 2 x 4 both yielding the result of 14
ORDER OF OPERATIONS
In order to solve any equation accurately it is imperative that one follow a very precise order of
the mathematical operations. Varying from the order will result in an inaccurate result to a given
equation.
The order of operations universally in mathematics is requires one to solve equations in the
following order working from left to right.
1st
Parenthesis
2nd Exponents
3rd Multiplication
4th Division
5th Addition
6th Subtraction
Many people find it easy to remember the order by its acronym P.E.M.D.A.S.
A simple example of the order can be found in the following example:
7 (42 + 2) – 8
In the order of operations we solve what is within the parenthesis first:
7 (16 + 2) – 8
Still solving the parenthesis in the first order we arrive at:
7 (18) - 8
being that the exponent was inside the parenthesis it falls into the first order of operations. Had
an exponent been outside the parenthesis it would now follow next in the order.
The order now comes to multiplying bringing us to:
126 – 8
This brings us to the final order of addition (not prevalent in this equation) and finally
subtraction:
126 subtracted by 8 gives us an answer of
118.
This conclusion can only be reached by adhering to the order of operations
WORD PROBLEMS
Word problems also have a guideline of an order to follow to give one the best opportunity of
figuring out the answer (this order is not required merely recommended).
1st Read the entire problem
2nd Re read the problem again looking for key words (i.e. “of”, “sum”, “between”)
3rd Use the keywords to plan a strategy using pictures, variables, graphs, etc.
4th Translate the words and problem into an equation
5th Solve the equation
6th look at the solution to the equation and be sure it makes sense to the context of the word
problem.
DECIMALS
Decimals are a method of writing fractional numbers without writing a fraction
having a numerator and denominator.
The fraction 7/10 could be written as the decimal 0.7
The period or decimal point indicates that this is a decimal.
The decimal 0.7 could be pronounced as SEVEN TENTHS or as ZERO POINT
SEVEN.
There are other decimals such as hundredths or thousandths. They all are based on
the number ten just like our number system.
A decimal may be greater than one. The decimal 3.7 would be pronounced as
THREE AND SEVEN TENTHS.
ADDING
How to add Decimals that have different numbers of decimal places
Write one number below the other so that the bottom decimal point is directly
below and lined up with the top decimal point.
Add each column starting at the right side.
Example: Add 3.2756 + 11.48
3.2756
11.48
14.7556
SUBTRACTING
Write the number that is being subtracted from. Write the number that is being
subtracted below the first number so that the decimal point of the bottom
number is directly below and lined up with the top decimal point.
Add zeros to the right side of the decimal with fewer decimal places so that
each decimal has the same number of decimal places.
Subtract the bottom number from the top number.
Example: Subtract 11.48 - 3.2756
11.4800
3.2756
8.2044
MULTIPLYING
Place one decimal above the other so that they are lined up on the right side.
Draw a line under the bottom number. Temporarily disregard the decimal
points and multiply the numbers like multiplying a three digit number by a
one digit number.
0.529
0.7
Multiply the two numbers on the right side. (9 * 7 = 63). This number is
larger than 10 so place a six above the center column and place three below
the line in the right column.
6
0.529
0.7
3
Multiply the digit in the top center column (2) by the digit in the center of the
right column (7). The answer (2*7=14) is added to the 6 above the center
column to give an answer of 20. The units place value (0) of 20 is placed
below the line and the tens place value (2) of the 20 is placed above the five.
26
0.529
0.7
03
The five of the top number is multiplied by the seven of the multiplier
(5*7=35). The two that was previously carried is added and 37 is placed
below the line. At the start we disregarded the decimal places. We must now
count up the decimal places and move the decimal place to its proper location.
We have three decimal places in 0.529 and one in the decimal 0.7 so we move
the decimal four places to the left to give the final answer of 0.3703.
26
0.529
0.7
0.3703
DIVISION
Place the divisor before the division bracket and place the dividend (0.4131)
under it.
0.17)0.4131
Multiply both the divisor and dividend by 100 so that the divisor is not a
decimal but a whole number. In other words move the decimal point two
places to the right in both the divisor and dividend
17)41.31
Proceed with the division as you normally would except put the decimal point
in the answer or quotient exactly above where it occurs in the dividend. For
example:
2.43
17)41.31
DECIMAL FRACTION CONVERSION
Decimals are a type of fractional number. The decimal 0.5 represents the fraction
5/10. The decimal 0.25 represents the fraction 25/100. Decimal fractions always
have a denominator based on a power of 10.
We know that 5/10 is equivalent to 1/2 since 1/2 times 5/5 is 5/10. Therefore, the
decimal 0.5 is equivalent to 1/2 or 2/4, etc.
Some common Equivalent Decimals and Fractions:
0.1 and 1/10
0.2 and 1/5
0.5 and 1/2
0.25 and 1/4
0.50 and 1/2
0.75 and 3/4
*
1.0 and 1/1
INTEGERS
Integers are numbers along a number line. They are both positive and negative in value.
.___.___.___.___.___.___.___.___.___.___.
-5 -4 -3 -2
-1
0
1
2
3
4
5
The number 0 represents the “origin” point where all values are derived from.
Every number on the number line has an “opposite”, meaning every negative number has
a positive opposite and vice versa, i.e. the opposite of 3 is -3
The opposite of a number is an equal in distance from the origin point on the number line
in the opposite direction.
The distance from the origin point a number has is it’s “absolute value”, i.e. the number 4
as well as the number –4 each are four away from the origin point of 0, thus making the
absolute value of both numbers 4
Adding and subtracting integers is almost one in the same…almost.
When adding two positive numbers one would simply use simple addition. When adding
a positive number and a negative number such as 4 + -3 one would just change the signs
in the equation so the calculation would derive from 4 – 3 with a result of 1
When adding two negative numbers such as –7 and –4 one uses the absolute value of the
number line to calculate the answer seven negatives plus four more negatives gives one a
total of eleven negatives. Thus the equation –7 – 4 yields a result of – 11
Multiplying and dividing integers employs a few simple rules to determine the value of
the conclusion:
In a two integer equation the result will always be negative if one of the computing
integers is negative
The result will always be positive if the two computing integers are of the same value
(either positive or negative)
In an equation with more than two integers, one counts the negative integers within the
equation. If the total amount of integers is an odd number then the result will always be
negative and always positive if the total amount is even.
GREATEST COMMON FACTOR
The greatest common factor is exactly what it represents in name. Going back to
factorization and the factor tree discussed earlier the greatest common factor would be
the largest number shared by two or more numbers that is a factor.
LEAST COMMON MULTIPLE
Again, exactly what it implies. The least common multiple is the lowest number two or
more numbers will divide into.
For example:
Multiples of the number 12 include but are not limited to 24, 36, 48, 60
Multiples of the number 20 include but are not limited to 40, 60, 80, 100
The “least common multiple” of these two numbers would be the number 60
FRACTIONS
In a fraction the number above the line is the numerator, the number below is the
denominator.
When multiplying fractions one multiplies the numerators together across the top and the
denominators across the bottom. When a conclusion is reached one then uses the least
common multiple to reduce the answer to its purest form.
For example:
2
____
3
3
X
____
8
would equal
6
___
24
because 6 is the
Greatest common
factor, the final answer would
reduce to: 1
___
4
Dividing fractions works in the same manner with an additional step: reversing the
numerator and denominator of the second fraction and then following the same process of
multiplication:
Adding fractions is a different process. Before any computation can be done one needs to
establish the least common multiple of the denominators involves then multiply the
numerators above them by the same amount:
For example:
6 / 8 + 18 / 36
The least common multiple of 8 and 36 is 72
8 is multiplied by 9 and 36 by 2
so now the number 6 is also multiplied by 9
and 18 by two. Now the equation reads as
54 / 72 + 36 / 72
No add together only the numerators leaving the common denominator as it is
Equating to 90 / 72
Dividing the denominator into the numerator to make sense of the conclusion is:
118 / 72
Reducing to its purest form leaves us with the final conclusion of:
11 / 4
Subtracting fractions works in the same process only subtracting the numerators rather
than adding them after the least common multiple has been established.
TRANSLATING AND SIMPLIFICATION
Simplifying an equation helps to insure an accurate solution. With longer equations one
can combine like terms to simplify the problem at hand. One needs to take into account
the rules of positive and negative, order of operations, distributive property, and
remember that variables with different exponents are not like terms.
Take into account this equation:
(3z2 + 4z – 7) + (7z2 - 5z – 8)
3z squared can be combined with 7z squared, 4z and –5z can also be combined and –7
and –8 can be combined as well.
The simplified equation now reads:
10z2 - z – 15
and now much easier to solve and yield an accurate result.
Using the distributive property in multiplication also simplifies an equation
Take into account this problem:
(7a + 6) (3a – 4)
using the F.O.I.L. method (first, outer, inner, last) the first set of numbers is distributed to
the second set leaving the equation to look like this:
21a2 - 20a + 18a – 24
now combining like terms to simplify further leaves us with:
21a2 - 10a – 24
again a much easier equation to solve.
Simplifying by division is subtracting like terms of the denominator from the numerator.
For example:
X3 Y5 / X Y would simplify into X2 Y4
Simplifying can also be done after a word problem has been turned into a workable
equation.
For example:
The sum of three times a number, and the difference between that number and seven.
Reads as
3x + X – 7
combining like terms gives us:
4x – 7
Once again a much easier problem to solve.
FINDING VARIABLES
Finding the value of a variable is a process that ultimately leaves the variable on one side
of an equation with its value on the other. To do this one would add, subtract, multiply, or
divide a number or variable from one side by the opposite operation on the other.
For example:
8a – 14 = 4a + 12
first we will subtract 4a from both sides leaving us with:
4a – 14 = 12
now we will add (opposite) 14 to both sides leaving us with:
4a = 26
now to isolate the variable we will divide (opposite operation) 4 from both sides, leaving
us with:
a = 61/2
so now we know that a is equal to 6 ½
INTRODUCTION TO METRIC MEASUREMENT
The metric system of measurement is based on factors of 10 from a base point or origin
of 0. Each point of measure has an appropriate prefix for the position of the decimal point
and an appropriate suffix for the appropriate form of measure.
Suffixes of measure are:
Liters to measure capacity
Grams to measure mass
And
Celsius to measure temperature
Prefixes for decimal position are:
milli representing .001
centi representing .01
deci representing .1
the base point of zero would be in this position
deka representing 10
hecto representing 100
and kilo representing 1000
RATIO
A ratio is a comparison of two numbers and can be read as:
12/13 is the same as 12:13, or 12 to 13
UNIT RATE
Mostly used in the application for cost (among other things however) unit rate is
simply the association of numbers.
For example:
$3.20 : 4oz = $.80 per ounce
PROPORTIONS
Proportions are just the comparison of two ratios or rates.
For example:
3/4 = 9/12 is a “true proportion” (3 x 12 = 36 as well as 9 x 4 = 36)
3/18 = 4/19 is not a true proportion (18 x 4 = 72 and 19 x 3 = 57)
The preceding was an overview of the first half of Modern College Mathematics
as taught at Westmoreland County Community College. This page is meant to
compliment the classroom material as a study tool for the first scheduled test.
*Source cited www.aaamath.com/dec
**Source cited WCCC
MTH 161 – 26 classroom