Find the HCF for these sets of nu a) 21 and 84 c) 30,210 and 700

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Sure, let's find the Highest Common Factor (HCF) for each set of numbers you've provided.

### a) HCF of 21 and 84

**Method 1: Prime Factorization**

1. **Prime factors of 21:**
   - $$21 = 3 	imes 7$$

2. **Prime factors of 84:**
   - $$84 = 2 	imes 2 	imes 3 	imes 7$$ (or $$2^2 	imes 3 	imes 7$$)

3. **Common prime factors:**
   - Both 21 and 84 have the prime factors **3** and **7**.

4. **HCF:**
   - Multiply the common prime factors: $$3 	imes 7 = 21$$

**Method 2: Division Approach**

- $$84 \div 21 = 4$$ with no remainder. 
- Since 21 divides 84 exactly, the HCF is **21**.

**Answer:** The HCF of 21 and 84 is **21**.

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### c) HCF of 30, 210, and 700

**Step 1: Prime Factorization**

1. **Prime factors of 30:**
   - $$30 = 2 	imes 3 	imes 5$$

2. **Prime factors of 210:**
   - $$210 = 2 	imes 3 	imes 5 	imes 7$$

3. **Prime factors of 700:**
   - $$700 = 2 	imes 2 	imes 5 	imes 5 	imes 7$$ (or $$2^2 	imes 5^2 	imes 7$$)

**Step 2: Identify Common Prime Factors**

- **2:** 
  - 30 has $$2^1$$
  - 210 has $$2^1$$
  - 700 has $$2^2$$
  - **Common:** $$2^1$$

- **3:** 
  - 30 has $$3^1$$
  - 210 has $$3^1$$
  - 700 has no factor of 3
  - **Common:** None

- **5:** 
  - 30 has $$5^1$$
  - 210 has $$5^1$$
  - 700 has $$5^2$$
  - **Common:** $$5^1$$

- **7:** 
  - 30 has no factor of 7
  - 210 has $$7^1$$
  - 700 has $$7^1$$
  - **Common:** None

**Step 3: Calculate HCF**

- Multiply the common prime factors with the lowest exponents:
  - $$2^1 	imes 5^1 = 2 	imes 5 = 10$$

**Answer:** The HCF of 30, 210, and 700 is **10**.

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### Summary

- **a)** HCF of **21** and **84** is **21**.
- **c)** HCF of **30**, **210**, and **700** is **10**.
The HCF of 21 and 84 is 21, and the HCF of 30, 210, and 700 is 10.

Find the HCF for these sets of nu
a) 21 and 84
c) 30,210 and 700

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