Determine the smallest number exactly divisible by 8, 15 and 24.

Find the least number which leaves a remainder 7 when divided by 18, 24 and 36.

Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 5.

Find the greatest number which exactly divides 27 and 33.

Find the largest number which exactly divides 48, 60 and 64

. Find the greatest number that exactly divides 368, 480 and 536.

Three bells toll at intervals of 9, 12 and 15 seconds respectively, beginning together. After what interval of time will they toll together?

Three athletes start running together, their steps measure 25 cm, 40 cm and 60 cm. If they are

running at the same speed, at what distance will they be taking steps together again? What is the largest number of students among whom 108 chocolates and 180 cookies can be equally distributed? How many chocolates and cookies will each student get?

Determine the length of the longest tape that can be used to measure exactly the dimensions of a hall of dimensions 4 m 25 cm, 5 m 50 cm and 6 m. [Hint: Convert m and cm into cm].

A boy saves 1.25 per day. What is the least number of days in which he will be able to convert his savings into a 50-rupee note?

Let's solve each question step-by-step:

### 1. Smallest number exactly divisible by 8, 15, and 24.

To find the smallest number exactly divisible by 8, 15, and 24, we need to find the Least Common Multiple (LCM) of the numbers.

**LCM of 8, 15, and 24:**

1. Prime factorization:
   - \( 8 = 2^3 \)
   - \( 15 = 3 \times 5 \)
   - \( 24 = 2^3 \times 3 \)

2. LCM is found by taking the highest power of each prime factor:
   - Highest power of 2: \( 2^3 \)
   - Highest power of 3: \( 3^1 \)
   - Highest power of 5: \( 5^1 \)

3. LCM = \( 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \).

**Answer:** The smallest number exactly divisible by 8, 15, and 24 is **120**.

### 2. Least number which leaves a remainder of 7 when divided by 18, 24, and 36.

We need to find the LCM of 18, 24, and 36 and then add 7.

**LCM of 18, 24, and 36:**

1. Prime factorization:
   - \( 18 = 2 \times 3^2 \)
   - \( 24 = 2^3 \times 3 \)
   - \( 36 = 2^2 \times 3^2 \)

2. LCM is found by taking the highest power of each prime factor:
   - Highest power of 2: \( 2^3 \)
   - Highest power of 3: \( 3^2 \)

3. LCM = \( 2^3 \times 3^2 = 8 \times 9 = 72 \).

Now, add 7: \( 72 + 7 = 79 \).

**Answer:** The least number which leaves a remainder of 7 when divided by 18, 24, and 36 is **79**.

### 3. Least number which when divided by 12, 16, 24, and 36 leaves a remainder of 5.

We need to find the LCM of 12, 16, 24, and 36 and then add 5.

**LCM of 12, 16, 24, and 36:**

1. Prime factorization:
   - \( 12 = 2^2 \times 3 \)
   - \( 16 = 2^4 \)
   - \( 24 = 2^3 \times 3 \)
   - \( 36 = 2^2 \times 3^2 \)

2. LCM is found by taking the highest power of each prime factor:
   - Highest power of 2: \( 2^4 \)
   - Highest power of 3: \( 3^2 \)

3. LCM = \( 2^4 \times 3^2 = 16 \times 9 = 144 \).

Now, add 5: \( 144 + 5 = 149 \).

**Answer:** The least number which when divided by 12, 16, 24, and 36 leaves a remainder of 5 is **149**.

### 4. Greatest number which exactly divides 27 and 33.

We need to find the Greatest Common Divisor (GCD) of 27 and 33.

1. Prime factorization:
   - \( 27 = 3^3 \)
   - \( 33 = 3 \times 11 \)

2. Common factor: \( 3^1 = 3 \).

**Answer:** The greatest number which exactly divides 27 and 33 is **3**.

### 5. Largest number which exactly divides 48, 60, and 64.

We need to find the GCD of 48, 60, and 64.

1. Prime factorization:
   - \( 48 = 2^4 \times 3 \)
   - \( 60 = 2^2 \times 3 \times 5 \)
   - \( 64 = 2^6 \)

2. Common factor: \( 2^2 = 4 \).

**Answer:** The largest number which exactly divides 48, 60, and 64 is **4**.

### 6. Greatest number that exactly divides 368, 480, and 536.

We need to find the GCD of 368, 480, and 536.

1. Prime factorization:
   - \( 368 = 2^4 \times 23 \)
   - \( 480 = 2^5 \times 3 \times 5 \)
   - \( 536 = 2^3 \times 67 \)

2. Common factor: \( 2^3 = 8 \).

**Answer:** The greatest number that exactly divides 368, 480, and 536 is **8**.

### 7. Interval of time after which three bells toll together.

We need to find the LCM of the intervals 9, 12, and 15 seconds.

**LCM of 9, 12, and 15:**

1. Prime factorization:
   - \( 9 = 3^2 \)
   - \( 12 = 2^2 \times 3 \)
   - \( 15 = 3 \times 5 \)

2. LCM = \( 2^2 \times 3^2 \times 5 = 180 \).

**Answer:** They will toll together every **180 seconds** (or 3 minutes).

### 8. Distance at which athletes take steps together.

We need to find the LCM of their step lengths: 25 cm, 40 cm, and 60 cm.

**LCM of 25, 40, and 60:**

1. Prime factorization:
   - \( 25 = 5^2 \)
   - \( 40 = 2^3 \times 5 \)
   - \( 60 = 2^2 \times 3 \times 5 \)

2. LCM = \( 2^3 \times 3 \times 5^2 = 600 \).

**Answer:** They will take steps together again at **600 cm**.

### 9. Largest number of students to equally distribute 108 chocolates and 180 cookies.

We need to find the GCD of 108 and 180.

1. Prime factorization:
   - \( 108 = 2^2 \times 3^3 \)
   - \( 180 = 2^2 \times 3^2 \times 5 \)

2. Common factor: \( 2^2 \times 3^2 = 36 \).

Each student gets:
- Chocolates: \( 108 \div 36 = 3 \).
- Cookies: \( 180 \div 36 = 5 \).

**Answer:** The largest number of students is **36**, with each getting **3 chocolates** and **5 cookies**.

### 10. Length of the longest tape to measure hall dimensions.

Dimensions in cm: 
- \( 4 \, m \, 25 \, cm = 425 \, cm \)
- \( 5 \, m \, 50 \, cm = 550 \, cm \)
- \( 6 \, m = 600 \, cm \).

Find the GCD of 425, 550, and 600.

1. Prime factorization:
   - \( 425 = 5^2 \times 17 \)
   - \( 550 = 2 \times 5^2 \times 11 \)
   - \( 600 = 2^3 \times 3 \times 5^2 \).

2. Common factor: \( 5^2 = 25 \).

**Answer:** The longest tape is **25 cm**.

### 11. Least number of days for savings to reach 50 rupees.

Daily saving: 1.25 rupees.

Days needed: \( 50 \div 1.25 = 40 \).

**Answer:** The least number of days is **40**.

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### Follow-up Questions:
1. What is the LCM and GCD, and how are they calculated?
2. Can you explain why the LCM is used in some problems and GCD in others?
3. How can prime factorization help in finding LCM and GCD?
4. What other methods can be used to find LCM and GCD besides prime factorization?
5. Why do you add remainders in specific problems involving LCM?

**Tip:** Always check for common prime factors to simplify the LCM and GCD calculations.

Explore solutions to math problems involving LCM (Least Common Multiple), GCD (Greatest Common Divisor), and remainders. Learn how to find the smallest number divisible by given integers, calculate when three bells toll together, and distribute chocolates and cookies equally. Suitable for Grades 6-8 students.

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