QA 54

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course mth 151

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/file3_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 4 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:

60

6*10

3*2*5*2

105

21*5

7*3*5

3*5=15

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):

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Self-critique Rating:

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Question: `q002. What are the prime factorizations of 84 and 126, and how can these two prime factorizations be used to find the greatest common divisor and the least common multiple of these two numbers?

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Your solution:

84

42*2

21*2*2

7*3*2*2

126

63*2

9*7*2

3*3*7*2

2*3*7=42

2*3*7*2*3-252

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):

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Self-critique Rating:

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Question: `q003. Find the greatest common divisor and least common multiple of 504 and 378.

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Your solution:

504

252*2

126*2*2

63*2*2*2

9*7*2*2*2

3*3*7*2*2*2

378

189*2

63*3*2

9*7*3*2

3*3*7*3*2

3*3*7*2=126

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

confidence rating #$&*:3

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Given Solution:

The prime factorizations are

504 = 2 * 2 * 2 * 3 * 3 * 7 and

378 = 2 * 3 * 3 * 3 * 7.

The greatest common divisor can contain just 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.

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Question: `q004. Find the greatest common factor and the least common divisor of 220 and 726.

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Your solution:

220

22*10

2*11*5*2

726

363*2

121*3*2

11*11*3*2

2*11=22

2*11*5*2*11*3=7260

confidence rating #$&*:

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Self-critique Rating:

@&

You don't say, which leaves the reader to guess what the various results mean.

I believe, however, that your conclusion is that 22 is the GCD and 7260 the LCM, and I can infer from your steps how you arrived at this conclusion.

*@

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Self-critique (if necessary):

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Self-critique rating:

&#Good responses. See my notes and let me know if you have questions. &#