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Prime Factorizations of 35, 41, 7










































Prime factorizations

Prime Factorization means to find all the prime factors of a number

   • A Prime Factor is a positive integer (or natural number):

             - greater than 1

             - cannot be divided by any positive integer other than itself and the number 1

             - The number 1 is not a prime factor.

   • Find the Prime factorizations of 35, 41, 7

Comments for Prime Factorizations of 35, 41, 7

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Oct 05, 2012
Prime Factorizations
by: Staff


Answer:


Part I


Prime factorization of 35 = 5*7

Prime factorization of 41 = 41

Prime factorization of 7 = 7

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To find the Prime Factorization of larger numbers, you may find a factor tree helpful.

For example:


Find the prime factors of 90


90 = 2 * 45


45 = 3 * 15


15 = 3 * 5



Prime Factors Tree – factors of 90







The Prime Factorization of 90 = 2 * 3 * 3 * 5



You can Factor an Algebraic Expression the same way.

For example:


Find the factors of 2x² + 10x + 12


2x² + 10x + 12 = 2 * (x² + 5x + 6)


x² + 5x + 6 = (x + 2) * (x + 3)


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Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part II

Factor Tree – Algebraic Expression




The Factors of:


2x² + 10x + 12 = 2 * (x + 2) * (x + 3)


Prime factorization breaks up numbers into their basic parts .

This allows you to:


Easily reduce factions to their lowest terms.


For Example, reduce the following fraction to its lowest terms:


                                294
                              -------
                               1470

Factor both the numerator and denominator.


294 = 2 * 3 *7 * 7


1470 = 2 * 3 * 5 * 7 * 7



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Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part III

Rewrite the faction in factored form


                                   2 * 3 *7 * 7
                              ----------------------
                                2 * 3 * 5 * 7 * 7

Cancel the common factors.
.

                                  2 * 3 * 7 * 7
                              ----------------------
                               2 * 3* 5 * 7 * 7


                                 1
                              ------
                                 5




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Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part IV

Reduce the following fraction containing algebraic expressions to its lowest terms:



                                a²bc³
                              ----------
                               a³b⁴c⁵

Factor both the numerator and denominator.


a²bc³ = a * a * b * c * c * c


a³b⁴c⁵ = a * a * a * b * b * b * b * c * c * c * c * c


Rewrite the faction in factored form


                        a * a * b * c * c * c
                         -------------------------------------------------------

      a * a * a * b * b * b * b * c * c * c * c * c


Cancel the common factors.



                      a * a * b * c * c * c
                           -------------------------------------------------------
    a * a * a * b * b * b * b * c * c * c * c * c


------------------------------------------

Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part V

                                             1
                              ---------------------------
                                a * b * b * b * c * c

                                   1
                              -----------
                                ab³c²

Simplify square roots and other roots
.

For example, simply the following square root:


                              √(882)

Factor the radicand.
.

882 = 2 * 3 * 3 * 7 * 7


Rewrite the square root in factored form


√(2 * 3 * 3 * 7 * 7)


Group the factors into squares.
.

√(2 * * )

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Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part VI

Remove squared numbers from under radical.
.

√( * ) * √(2)


3 * 7 * √(2)


21 * √(2)


21√(2)



Simplify the following square root which contains an algebraic expression:


                              √(a³b⁴c⁵)

Factor the radicand.
.

a³b⁴c⁵


= a * a * a * b * b * b * b* c * c* c * c * c


Rewrite the square root in factored form


√( a * a * a * b * b * b * b* c * c* c * c * c)


Group the factors into squares.
.

√( a * * * * * * c)


Remove squared numbers from under radical.
.

√( * * * * ) * √(a * c)


a * b * b * c * c * √(2)


a * * * √(2)

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Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part VII

ab²c²√(ac)


Factoring is used to find the lowest common denominator when adding and subtracting fractions


For Example, add the following two fractions:


                                 3             1
                              ------  +   ------
                                 8             6

Factoring is used to find the lowest common denominator:


Factor the denominator of each fraction.


8 = 2 * 2 * 2 = 2³


6 = 2 * 3


Lowest common denominator = 2³ * 3 = 24



Convert each fraction to the lowest common denominator.


                            3         3            1        4
                          ----- * -----  +   ----- * -----
                            8         3            6        4
------------------------------------------

Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part VIII

                                 9              4
                              ------  +   ------
                                24            24

Add the fractions.
.

                               (9 + 4)
                              -----------
                                  24

                                  13
                              -----------
                                  24

Add the following two fractions which contain algebraic expressions:


                                 a             c
                              ------  +   ------
                                dc           d²e
------------------------------------------

Oct 05, 2012
Prime Factorizations
by: Staff

------------------------------------------

Part IX

Factoring is used to find the lowest common denominator:


Factor the denominator of each fraction.
.

dc = d * c


d²e = d * d * e = d²e


Lowest common denominator


= d² * c * e = d²ce


Convert each fraction to the lowest common denominator.


                           a        de          c         c
                        ----- * -----  +   ----- * -----
                           dc      de         d²e       c


                               ade           c²
                              ------  +  -------
                              d²ce         d²ce

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Oct 05, 2012
Prime Factorizations
by: Staff


------------------------------------------

Part X

Add the fractions.
.

                               (ade + c²)
                              ----------------
                                  d²ce


Prime Factorization is also used in encryption
.

Many encryption algorithms use very, very large prime factors which are multiplied together to create a public encryption key.


The prime factors themselves are used to create the decryption key.


The RSA encryption algorithm, which is used to encrypt commerce web sites, makes use of this idea.


To determine RSA’s decryption key, the public key must be factored.


There is no easy way to factor a large public encryption key. Every possible combination of factors must be tried – one at a time. You must first try dividing by 2, then 3, then 4, and so on.


Even with a high speed computer, factoring a public encryption key of 1024 bits may take years, or centuries.







Thanks for writing.

Staff
www.solving-math-problems.com


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