a) List the prime numbers up to 40. $$ \begin{array}{lll} ext { b) } 1557 ext { a prime number? Explain. } \ ext { nd the HCF and LCM for these sets of numbers using prime factor } \ 40 ext { and } 56 & ext { b) } 60 ext { and } 150 & ext { c) } 36 ext { and } 84 .\end{array} $$

Question

a) List the prime numbers up to 40. $$ \begin{array}{lll}	ext { b) } 1557 	ext { a prime number? Explain. } \ 	ext { nd the HCF and LCM for these sets of numbers using prime factor } \ 40 	ext { and } 56 & 	ext { b) } 60 	ext { and } 150 & 	ext { c) } 36 	ext { and } 84 .\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's tackle each part of your question step by step.

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### **a) List the prime numbers up to 40.**

**Prime Numbers** are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Here are all the prime numbers up to 40:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37

---

### **b) Is 1557 a prime number? Explain.**

**No, 1557 is not a prime number.**

**Explanation:**

To determine whether 1557 is a prime number, we need to check if it has any divisors other than 1 and itself.

1. **Check Divisibility by Small Primes:**

- **Divisible by 2:** 1557 is odd, so it's not divisible by 2.
   
   - **Divisible by 3:** 
     - Sum of digits = 1 + 5 + 5 + 7 = 18
     - 18 is divisible by 3, so 1557 is divisible by 3.
   
   - **Calculation:**
     - 1557 ÷ 3 = 519
     - Therefore, 1557 = 3 × 519

Since 1557 can be divided by 3 (and hence by other numbers like 519), it has divisors other than 1 and itself, confirming that it is **not a prime number**.

---

### **c) Find the HCF and LCM for the following sets of numbers using prime factorization.**

To find the **Highest Common Factor (HCF)** and **Least Common Multiple (LCM)** using prime factorization, follow these steps:

1. **Find the prime factors of each number.**
2. **For HCF:** Multiply the lowest powers of common prime factors.
3. **For LCM:** Multiply the highest powers of all prime factors present.

Let's apply this to each pair:

---

#### **i) 40 and 56**

1. **Prime Factorization:**
   - **40:**
     - 40 ÷ 2 = 20
     - 20 ÷ 2 = 10
     - 10 ÷ 2 = 5
     - 5 ÷ 5 = 1
     - **Prime Factors:** $$ 2^3 	imes 5 $$
   
   - **56:**
     - 56 ÷ 2 = 28
     - 28 ÷ 2 = 14
     - 14 ÷ 2 = 7
     - 7 ÷ 7 = 1
     - **Prime Factors:** $$ 2^3 	imes 7 $$

2. **HCF:**
   - Common prime factors: $$ 2^3 $$
   - **HCF = $$ 2^3 = 8 $$**

3. **LCM:**
   - Combine all prime factors with the highest powers:
   - $$ 2^3 	imes 5 	imes 7 = 8 	imes 5 	imes 7 = 280 $$
   - **LCM = 280**

---

#### **ii) 60 and 150**

1. **Prime Factorization:**
   - **60:**
     - 60 ÷ 2 = 30
     - 30 ÷ 2 = 15
     - 15 ÷ 3 = 5
     - 5 ÷ 5 = 1
     - **Prime Factors:** $$ 2^2 	imes 3 	imes 5 $$
   
   - **150:**
     - 150 ÷ 2 = 75
     - 75 ÷ 3 = 25
     - 25 ÷ 5 = 5
     - 5 ÷ 5 = 1
     - **Prime Factors:** $$ 2 	imes 3 	imes 5^2 $$

2. **HCF:**
   - Common prime factors: $$ 2^1 	imes 3^1 	imes 5^1 $$
   - **HCF = $$ 2 	imes 3 	imes 5 = 30 $$**

3. **LCM:**
   - Combine all prime factors with the highest powers:
   - $$ 2^2 	imes 3 	imes 5^2 = 4 	imes 3 	imes 25 = 300 $$
   - **LCM = 300**

---

#### **iii) 36 and 84**

1. **Prime Factorization:**
   - **36:**
     - 36 ÷ 2 = 18
     - 18 ÷ 2 = 9
     - 9 ÷ 3 = 3
     - 3 ÷ 3 = 1
     - **Prime Factors:** $$ 2^2 	imes 3^2 $$
   
   - **84:**
     - 84 ÷ 2 = 42
     - 42 ÷ 2 = 21
     - 21 ÷ 3 = 7
     - 7 ÷ 7 = 1
     - **Prime Factors:** $$ 2^2 	imes 3 	imes 7 $$

2. **HCF:**
   - Common prime factors: $$ 2^2 	imes 3^1 $$
   - **HCF = $$ 4 	imes 3 = 12 $$**

3. **LCM:**
   - Combine all prime factors with the highest powers:
   - $$ 2^2 	imes 3^2 	imes 7 = 4 	imes 9 	imes 7 = 252 $$
   - **LCM = 252**

---

### **Summary**

| **Pair**   | **HCF** | **LCM** |
|------------|---------|---------|
| 40 & 56    | 8       | 280     |
| 60 & 150   | 30      | 300     |
| 36 & 84    | 12      | 252     |

---

I hope this helps! If you have any more questions, feel free to ask.
**Prime Numbers up to 40:** 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 **1557 is not a prime number** because it is divisible by 3. **HCF and LCM:** - **40 and 56:** - HCF: 8 - LCM: 280 - **60 and 150:** - HCF: 30 - LCM: 300 - **36 and 84:** - HCF: 12 - LCM: 252

a) List the prime numbers up to 40. \( \begin{array}{lll}\text { b) } 1557 \text { a prime number? Explain. } \\ \text { nd the HCF and LCM for these sets of numbers using prime factor } \\ 40 \text { and } 56 & \text { b) } 60 \text { and } 150 & \text { c) } 36 \text { and } 84 .\end{array} \)

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