MAT 126 Survey of Mathematical - LCM and GCF

List the ages of two people in your life, one older than you and one younger than you. It would be best if the younger person was 15 years of age or younger.
Find the prime factorizations of your age and the other two persons’ ages. Show your work listed by name and age. Make sure your work is clear and concise.
Find the LCM and the GCF for each set of numbers. Again, be clear and concise. Explain or show how you arrived at your answers.
In your own words, explain the meaning of your calculated LCM and GCF for the ages you selected. Do not explain how you got the numbers; rather explain the meaning of the numbers. Be specific to your numbers; do not give generic definitions.
Respond to at least two of your classmates’ postings. Did your classmates calculate the LCM and GCF correctly? Are their interpretations correctly applied to the ages?
Your initial

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Oct 27, 2011

LCM and GCF
by: Staff

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Part II

There is only one number which appears as one of the prime numbers for all three ages: 5

How many times is the number 5 listed for each age?

Your Age: 25, prime factors = 5 * 5
5 is listed twice

Younger Person: 10, prime factors = 5 * 2
5 is listed once

Older Person: 45, prime factors = 5 * 9
5 is listed once

Pick the lowest number of times five is listed.

The number 5 is listed one time as a prime factor for all three ages. (Although 5 is listed twice as a prime factor for the number 25, it is not listed twice for 10 or 45.)

GCF = 5

In your own words, explain the meaning of your calculated LCM and GCF for the ages you selected.

LCM: The least common multiple of two or more numbers is three things: 1) first, a MULTIPLE; 2) second, a COMMON MULTIPLE; and 3) last, the LEAST OF ALL the possible COMMON MULTIPLES.

There are many multiples for every number.

There will be many common multiples for any two (or more) numbers.

But, . . . there will be only one LCM.

For example, some of the multiples of 2 are:

2 * 1 = 2
2 * 2 = 4
2 * 3 = 6
2 * 4 = 8
2 * 5 = 10
2 * 6 = 12
2 * 7 = 14
2 * 8 = 16
2 * 9 = 18
.
.
.

Some of the multiples of 3 are:

3 * 1 = 3
3 * 2 = 6
3 * 3 = 9
3 * 4 = 12
3 * 5 = 15
3 * 6 = 18
3 * 7 = 21
.
.
.

In this example, the multiples of 2 which are listed are:

2
4
6
8
10
12
14
16
18
.
.
.

And the multiples of 3 which are listed are:

3
6
9
12
15
18
21
.
.
.

Some of the multiples of 2 are the same as some of the multiples of 3.

The multiples of 6, 12, and 18 appear in both lists.

6, 12, and 18 are some of the COMMON MULTIPLES for the numbers 2 and 3

The LCM is the smallest of the common multiples (the smallest of the numbers 6, 12, and 18).

LCM = 6

GCF: The greatest common factor of two or more numbers is three things: 1) first, a FACTOR; 2) second, a COMMON FACTOR; and 3) last, the GREATEST OF ALL the COMMON FACTORS.

There may be many factors for every number.

There may be many common factors for any two (or more) numbers.

But, . . . there will be only one GCF.

For example, factors of 48 are:

2 * 2 * 2 * 2 * 3 = 48

The factors of 32 are:

2 * 2 * 2 * 2 * 2 = 32

So, the factors of 48 are can be written:

48 = 2⁴ * 3

48 = 16 * 3

And the factors of 32 can be written:

32 = 2⁴ * 2

32 = 16 * 2

The number 16 is a factor.

The number 16 is a common factor (not a prime factor). It appears as a factor for both 48 and 32.

16 is the largest common factor which appears as a factor for both 48 and 32.

GCF = 16

Thanks for writing.

Staff
www.solving-math-problems.com

Oct 27, 2011

LCM and GCF
by: Staff

Part I

Question:

List the ages of two people in your life, one older than you and one younger than you. It would be best if the younger person was 15 years of age or younger.
Find the prime factorizations of your age and the other two persons’ ages. Show your work listed by name and age. Make sure your work is clear and concise.
Find the LCM and the GCF for each set of numbers. Again, be clear and concise. Explain or show how you arrived at your answers.
In your own words, explain the meaning of your calculated LCM and GCF for the ages you selected. Do not explain how you got the numbers; rather explain the meaning of the numbers. Be specific to your numbers; do not give generic definitions.
Respond to at least two of your classmates’ postings. Did your classmates calculate the LCM and GCF correctly? Are their interpretations correctly applied to the ages?
Your initial

Answer:

List the ages of two people in your life, one older than you and one younger than you. It would be best if the younger person was 15 years of age or younger.

You did not list any ages, so I’m going to arbitrarily pick some.

Your Age: 25
Younger Person: 10
Older Person: 45

Find the prime factorizations of your age and the other two persons’ ages. Show your work listed by name and age.

Your Age: 25, prime factors = 5 * 5

Younger Person: 10, prime factors = 2 * 5

Older Person: 45, prime factors = 5 * 9

Find the LCM and the GCF for each set of numbers. Explain or show how you arrived at your answers.

You can find the LCM (least common multiple) using the prime factors which you have already computed

List the prime factors again, being careful to show the exponents associated with each:

Your Age: 25, prime factors = 5 * 5 = 5²

Younger Person: 10, prime factors = 5 * 2 = 5¹ * 2¹

Older Person: 45, prime factors = 5 * 9 = 5¹ * 9¹

All together, there are the only three prime factors listed: 5; 2; and 9

To compute the LCM:

Choose the 5 with the highest exponent: 5²
(this comes from Your Age)

Choose the 2 with the highest exponent: 2¹
(this comes from the Younger Person’s Age)

Choose the 9 with the highest exponent: 9¹
(this comes from the Older Person’s Age)

Multiply all three of these numbers together

LCM = 5² * 2¹ * 9¹ = 25 * 2 * 9 = 450

LCM (least common multiple) for the values 25, 10, 45 = 450

You can also find the GCF (greatest common factor) using the same prime factors which you used to compute the LCM.

Your Age: 25, prime factors = 5 * 5

Younger Person: 10, prime factors = 5 * 2

Older Person: 45, prime factors = 5 * 9

Which factors appear as a prime factor for all three ages?

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MAT 126 Survey of Mathematical - LCM and GCF

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