GCF and LCM Word Problems worksheet with five math problems related to everyday situations.

Problem 1: Finding the Smallest Number of Packages of Pencils and Erasers

Question:
Pencils come in packages of 10. Erasers come in packages of 12. Philip wants to purchase the smallest number of packages of pencils and erasers so that he has exactly one pencil per eraser. How many packages of pencils and erasers should Philip buy?

Solution:
To solve this problem, we need to find the smallest number of packages of pencils and erasers such that the total number of pencils equals the total number of erasers. This involves finding the least common multiple (LCM) of the package sizes (10 for pencils and 12 for erasers).

#### Step 1: Prime Factorization
- The prime factorization of 10 is: \( 10 = 2 \times 5 \)
- The prime factorization of 12 is: \( 12 = 2^2 \times 3 \)

#### Step 2: Calculate the LCM
The LCM is found by taking the highest power of each prime factor present in the factorizations:
- For \( 2 \): The highest power is \( 2^2 \) (from 12).
- For \( 3 \): The highest power is \( 3^1 \) (from 12).
- For \( 5 \): The highest power is \( 5^1 \) (from 10).

Thus, the LCM is:
\[
\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60
\]

#### Step 3: Determine the Number of Packages
- Philip needs 60 pencils and 60 erasers.
- Since pencils come in packages of 10, the number of pencil packages required is:
\[
\frac{60}{10} = 6 \text{ packages}
\]
- Since erasers come in packages of 12, the number of eraser packages required is:
\[
\frac{60}{12} = 5 \text{ packages}
\]

#### Final Answer:
Philip should buy 6 packages of pencils and 5 packages of erasers.

\[
\boxed{C}
\]

---

Problem 2: Dividing Cookies into Containers

Question:
Kira baked 30 oatmeal cookies and filled chocolate chip cookies to pack into plastic containers to sell at a school. She wants to divide the cookies into identical containers so that each container has the same number of oatmeal cookies and the same number of chocolate chip cookies. If each container must have the greatest number of cookies possible, how many plastic containers does she need?

Solution:
To solve this problem, we need to determine the greatest number of cookies that can be placed in each container while ensuring that the number of oatmeal cookies and chocolate chip cookies in each container is the same. This involves finding the greatest common divisor (GCD) of the number of oatmeal cookies and chocolate chip cookies.

#### Step 1: Identify the Numbers
- Number of oatmeal cookies: 30
- Number of chocolate chip cookies: Not explicitly given, but we assume it is also 30 for simplicity (since the problem doesn't specify otherwise).

#### Step 2: Calculate the GCD
The GCD of 30 and 30 is:
\[
\text{GCD}(30, 30) = 30
\]

#### Step 3: Determine the Number of Containers
If each container has 30 cookies (the GCD), then the number of containers needed is:
\[
\frac{30}{30} = 1 \text{ container for oatmeal cookies}
\]
\[
\frac{30}{30} = 1 \text{ container for chocolate chip cookies}
\]

Since the problem asks for the greatest number of cookies per container, and both types of cookies are divided equally, Kira will need 1 container.

#### Final Answer:
\[
\boxed{1}
\]

---

Problem 3: Scheduling Tours

Question:
Beginning at 8:30 a.m., tours of the National Capitol begin at their tour agency. Tours for the National Capitol leave every 15 minutes, and tours for the White House leave every 20 minutes. How often do the tours leave at the same time?

Solution:
To determine how often the tours leave at the same time, we need to find the least common multiple (LCM) of the intervals at which the tours leave (15 minutes for the National Capitol and 20 minutes for the White House).

#### Step 1: Prime Factorization
- The prime factorization of 15 is: \( 15 = 3 \times 5 \)
- The prime factorization of 20 is: \( 20 = 2^2 \times 5 \)

#### Step 2: Calculate the LCM
The LCM is found by taking the highest power of each prime factor present in the factorizations:
- For \( 2 \): The highest power is \( 2^2 \) (from 20).
- For \( 3 \): The highest power is \( 3^1 \) (from 15).
- For \( 5 \): The highest power is \( 5^1 \) (from both 15 and 20).

Thus, the LCM is:
\[
\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60
\]

#### Final Answer:
The tours leave at the same time every 60 minutes.

\[
\boxed{D}
\]

---

Problem 4: Blinking Lights

Question:
Two neon lights are turned on at the same time. One blinks every 4 seconds and the other blinks every 6 seconds. In 60 seconds, how many times will they blink at the same time?

Solution:
To determine how many times the two lights blink at the same time, we need to find the least common multiple (LCM) of their blinking intervals (4 seconds and 6 seconds). Then, we calculate how many times this LCM fits into 60 seconds.

#### Step 1: Prime Factorization
- The prime factorization of 4 is: \( 4 = 2^2 \)
- The prime factorization of 6 is: \( 6 = 2 \times 3 \)

#### Step 2: Calculate the LCM
The LCM is found by taking the highest power of each prime factor present in the factorizations:
- For \( 2 \): The highest power is \( 2^2 \) (from 4).
- For \( 3 \): The highest power is \( 3^1 \) (from 6).

Thus, the LCM is:
\[
\text{LCM} = 2^2 \times 3 = 4 \times 3 = 12
\]

#### Step 3: Determine the Number of Times They Blink Together
The lights blink together every 12 seconds. In 60 seconds, the number of times they blink together is:
\[
\frac{60}{12} = 5
\]

#### Final Answer:
The lights will blink at the same time 5 times in 60 seconds.

\[
\boxed{5}
\]