Math - Age Problem

by Aleshia Hunter
(Logan, West Virginia, United States)

Demonstrate your understanding of the following terms by completing the exercise explained below:

   • Prime Factorizations

   • Least Common Multiple (LCM)

   • Greatest Common Factor (GCF)

List:

  1. Your name and age

  2. The names and ages of two people you know

    a) one person should be older than you

    b) one person should be younger than you
       (if you can, choose a person that is
       at least 15 years your junior)

Find:

  1. The prime factorization of each of the three ages
       (Show the results by name and age)

Together, the three ages form a “set” of three numbers

  2. Find the LCM for the set
       (explain how you determined the LCM)

  3. Find the GCF for the set
       (explain how you determined the GCF)

In your own words, explain the following two concepts as they relate to the three ages. Explain the meaning of your calculations, not how you arrived at the numbers.

  1. LCM

  2. GCF

Ages Selected:

   • Myself age 40
   • Husband age 52
   • nephew age 13

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Jan 04, 2012

LCM and GCF
by: Staff

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Part III

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

Jan 04, 2012

LCM and GCF
by: Staff

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

Part II

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

Myself age: 40, factors = 1 * 2 * 2 * 2 * 5

Husband age: 52, factors = 1 * 2 * 2 * 13

Nephew age: 13, factors = 1 * 13

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

There is only one number which appears as one of the prime numbers for all three ages: 1

GCF (for the numbers 40, 52, and 13) = 1

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Parenthetical Note:

The particular ages you selected don’t provide a good, general example which can be used to demonstrate how to find the GCF

To demonstrate the process for finding the GCF, I have chosen three other numbers: 10, 75, and 20

10, factors = 2 * 5 = 2¹ * 5¹

75, factors = 3 * 5 * 5 = 3¹ * 5²

20, factors = 2 * 2 * 5 = 2² * 5¹

How many times is the number 2 listed for each number?

First Number: 10, prime factors = 2¹ * 5¹
2 is listed 1 time

Second Number: 75, prime factors = 3¹ * 5²
2 is listed 0 (zero) times

Third Number: 20, prime factors = 2² * 5¹
2 is listed 2 times

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

First Number: 10, prime factors = 2¹ * 5¹
5 is listed 1 time

Second Number: 75, prime factors = 3¹ * 5²
5 is listed 2 times

Third Number: 20, prime factors = 2² * 5¹
5 is listed 1 time

How many times is the number 3 listed for each number?

First Number: 10, prime factors = 2¹ * 5¹
3 is listed 0 (zero) times

Second Number: 75, prime factors = 3¹ * 5²
3 is listed 1 time

Third Number: 20, prime factors = 2² * 5¹
3 is listed 0 (zero) times

Since the 2 and the 3 are not listed as a factor for all three numbers, they are NOT common factors.

ONLY 5 is a common factor

Pick the lowest number of times five is listed.
The number 5 is listed one time as a prime factor for all three numbers. (Although 5 is listed twice as a prime factor for the number 75, it is not listed twice for 10 or 20.)

GCF (for the numbers 10, 75, and 20) = 5

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

Jan 04, 2012

LCM and GCF
by: Staff

Part I

Question:

by Aleshia Hunter
(Logan, West Virginia, United States)

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.

Myself age 40
Husband age 52
nephew age 13

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.

Myself age: 40

Husband age: 52

Nephew age: 13

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

Myself age: 40, prime factors = 2 * 2 * 2 * 5

Husband age: 52, prime factors = 2 * 2 * 13

Nephew age: 13, prime factors = 13

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:

Myself age: 40, prime factors = 2 * 2 * 2 * 5 = 2³ * 5¹

Husband age: 52, prime factors = 2 * 2 * 13 = 2² * 13¹

Nephew age: 13, prime factors = 13 = 13¹

All together, there are the only three prime factors listed: 2, 5, and 13

To compute the LCM:

Choose the 2 with the highest exponent: 2³
(this comes from Myself age)

Choose the 5 with the highest exponent: 5¹
(this also comes from Myself age)

Choose the 13 with the highest exponent: 13¹
(this comes from either Husband age, or Nephew age)

Multiply all three of these numbers together

LCM = 2³ * 5¹ * 13¹ = 8 * 5 * 13 = 520

LCM (least common multiple) for the values (ages) of 40, 52, and 13 = 520

=========================

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Math - Age Problem

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