025 GCFLCM

#$&*

course Mth 151

4/30 10

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file

3_259.htm

.

Your solution, attempt at solution. If you are unable to attempt a solution, give a

phrase-by-phrase interpretation of the problem along with a statement of what you do

or do not understand about it. This response should be given, based on the work you

did in completing the assignment, before you look at the given solution.

025. GCF, LCM

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Question: `q001. There are three questions in this assignment.

2 * 2 * 3 * 5 = 60 and 3 * 5 * 7 = 105.

What do the prime factorizations of 60 and 105 having common?

What is the prime factorization of the smallest number which contains within its

prime factorization the prime factorizations of both 60 and 105?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

3*5=15 GCD

2*2*3*5*7=420 LCM

confidence rating #$&*:3

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Given Solution:

The prime factorizations 2 * 2 * 3 * 5 = 60 and 3 * 5 * 7 = 105 have in common the

product 3 * 5 = 15. This is the largest number that will divide evenly into both 60

and 105, and is called the greatest common divisor of 60 and 105.

In order to contain to both of the prime factorizations 2 * 2 * 3 * 5 = 60 and 3 * 5

* 7 = 105 a number must contain in its prime factorizations the entire prime

factorization 2 * 2 * 3 * 5, and in addition the 7 still necessary in order to

contain 3 * 5 * 7. Thus the number must be 2 * 2 * 3 * 5 * 7 = 420. This number is a

multiple of both 2 * 2 * 3 * 5 = 60 and 3 * 5 * 7 = 120, and is the smallest number

which is a multiple of both. We therefore call 420 the Least Common Multiple of 60

and 105.

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Self-critique (if necessary):OK

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Self-critique Rating:3

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Question: `q002. What are the prime factorizations of 84 and 126, and how can they

be used to find the greatest common divisor and the least common multiple of these

two numbers?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

84 2*2*3*7

126 2*3*3*7

2*3*7=42 GCD

2*2*3*3*7=252 LCM

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

The prime factorization of 84 is 2 * 2 * 3 * 7, and the prime factorization of 126

is 2 * 3 * 3 * 7.

The greatest common divisor of these numbers is the number we build up from all the

primes that are common to both of these prime factorizations. The two prime

factorizations having common 2, 3 and 7, which give us the greatest common divisor 2

* 3 * 7 = 42.

The least common multiple is made up of just those primes which are absolutely

necessary to contain the two given numbers. This number would have to contain the

first number 2 * 2 * 3 * 7, and would in addition need another 3 in order to contain

2 * 3 * 3 * 7. The least common multiple is therefore 2 * 2 * 3 * 3 * 7 = 252.

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Self-critique (if necessary):OK

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Self-critique Rating:3

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Question: `q003. Find the greatest common divisor and least common multiple of 504

and 378.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

504 2*2*2*3*3*7

378 2*3*3*3*7

2*3*3*7=126 GCD

2*2*2*3*3*3*7=3024

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

We find that 504 = 2 * 2 * 2 * 3 * 3 * 7 and 378 = 2 * 3 * 3 * 3 * 7. The greatest

common divisor can contain a single 2 since 378 has only a single 2 in its

factorization, two 3's since both numbers contain at least two 3's, and a single 7.

The greatest common divisor is therefore 2 * 3 * 3 * 7 = 126.

The least common multiple must contain the first number, 2 * 2 * 2 * 3 * 3 * 7, and

another 3 because of the third 3 in 378. The least common multiple is therefore 2 *

2 * 2 * 3 * 3 * 3 * 7 = 1512.

"

Self-critique (if necessary):

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

Self-critique rating:

*********************************************

Question: `q003. Find the greatest common divisor and least common multiple of 504

and 378.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

504 2*2*2*3*3*7

378 2*3*3*3*7

2*3*3*7=126 GCD

2*2*2*3*3*3*7=3024

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

We find that 504 = 2 * 2 * 2 * 3 * 3 * 7 and 378 = 2 * 3 * 3 * 3 * 7. The greatest

common divisor can contain a single 2 since 378 has only a single 2 in its

factorization, two 3's since both numbers contain at least two 3's, and a single 7.

The greatest common divisor is therefore 2 * 3 * 3 * 7 = 126.

The least common multiple must contain the first number, 2 * 2 * 2 * 3 * 3 * 7, and

another 3 because of the third 3 in 378. The least common multiple is therefore 2 *

2 * 2 * 3 * 3 * 3 * 7 = 1512.

"

Self-critique (if necessary):

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Self-critique rating:

#*&!

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