4TH GRADE MATH - GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE ... - Free Printable

Problem Analysis and Solutions

The provided document contains four word problems that involve the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM). Below, I will solve each problem step by step with clear explanations.

---

Problem 1:

Sara has 16 red flowers and 24 yellow flowers. She wants to make bouquets with the same number of each color flower in each bouquet. What is the greatest number of bouquets she can make?

#### Solution:
To determine the greatest number of bouquets Sara can make, we need to find the Greatest Common Factor (GCF) of the number of red flowers (16) and yellow flowers (24). The GCF will tell us the largest number of bouquets that can be made such that each bouquet has the same number of red and yellow flowers.

1. Find the factors of 16:
\[
16 = 1, 2, 4, 8, 16
\]

2. Find the factors of 24:
\[
24 = 1, 2, 3, 4, 6, 8, 12, 24
\]

3. Identify the common factors:
\[
\text{Common factors of 16 and 24 are: } 1, 2, 4, 8
\]

4. Determine the greatest common factor:
\[
\text{GCF}(16, 24) = 8
\]

5. Interpret the result:
Sara can make 8 bouquets, with each bouquet containing:
- \( \frac{16}{8} = 2 \) red flowers
- \( \frac{24}{8} = 3 \) yellow flowers

#### Final Answer:
\[
\boxed{8}
\]

---

Problem 2:

Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every 15 seconds. In how many seconds will they blink together again?

#### Solution:
To determine when the two neon signs will blink together again, we need to find the Least Common Multiple (LCM) of their blinking intervals (9 seconds and 15 seconds). The LCM will give us the smallest time interval after which both signs will blink simultaneously.

1. Prime factorization of 9:
\[
9 = 3^2
\]

2. Prime factorization of 15:
\[
15 = 3 \times 5
\]

3. Find the LCM:
The LCM is obtained by taking the highest power of each prime factor present in the factorizations:
\[
\text{LCM}(9, 15) = 3^2 \times 5 = 9 \times 5 = 45
\]

4. Interpret the result:
The two neon signs will blink together again after 45 seconds.

#### Final Answer:
\[
\boxed{45}
\]

---

Problem 3:

Lisa is making activity baskets to donate to charity. She has 12 coloring books, 28 markers, and 36 crayons. What is the greatest number of baskets she can make if each type of toy is equally distributed among the baskets? How many of each supply will go into the baskets?

#### Solution:
To determine the greatest number of baskets Lisa can make, we need to find the Greatest Common Factor (GCF) of the quantities of coloring books (12), markers (28), and crayons (36). The GCF will tell us the largest number of baskets that can be made such that each basket has an equal number of each type of supply.

1. Find the factors of 12:
\[
12 = 1, 2, 3, 4, 6, 12
\]

2. Find the factors of 28:
\[
28 = 1, 2, 4, 7, 14, 28
\]

3. Find the factors of 36:
\[
36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
\]

4. Identify the common factors:
\[
\text{Common factors of 12, 28, and 36 are: } 1, 2, 4
\]

5. Determine the greatest common factor:
\[
\text{GCF}(12, 28, 36) = 4
\]

6. Calculate the number of each supply per basket:
- Coloring books per basket: \( \frac{12}{4} = 3 \)
- Markers per basket: \( \frac{28}{4} = 7 \)
- Crayons per basket: \( \frac{36}{4} = 9 \)

7. Interpret the result:
Lisa can make 4 baskets, with each basket containing:
- 3 coloring books
- 7 markers
- 9 crayons

#### Final Answer:
\[
\boxed{4}
\]

---

Problem 4:

The school cafeteria serves tacos every sixth day and cheeseburgers every eighth day. If tacos and cheeseburgers are both on today's menu, how many days will it be before they are both on the menu again?

#### Solution:
To determine when tacos and cheeseburgers will both be on the menu again, we need to find the Least Common Multiple (LCM) of the serving intervals (6 days for tacos and 8 days for cheeseburgers). The LCM will give us the smallest number of days after which both items will be served on the same day.

1. Prime factorization of 6:
\[
6 = 2 \times 3
\]

2. Prime factorization of 8:
\[
8 = 2^3
\]

3. Find the LCM:
The LCM is obtained by taking the highest power of each prime factor present in the factorizations:
\[
\text{LCM}(6, 8) = 2^3 \times 3 = 8 \times 3 = 24
\]

4. Interpret the result:
Tacos and cheeseburgers will both be on the menu again after 24 days.

#### Final Answer:
\[
\boxed{24}
\]

---

Summary of Answers:

1. \(\boxed{8}\)
2. \(\boxed{45}\)
3. \(\boxed{4}\)
4. \(\boxed{24}\)