4. Find a common prime factor in each of these pairs of numbers. $$ \begin{array}{lll} ext { a) } 21 ext { and } 411 & ext { b) } 77 ext { and } 242 & ext { c) } 56 ext { and } 63\end{array} $$

Question

4. Find a common prime factor in each of these pairs of numbers. $$ \begin{array}{lll}	ext { a) } 21 	ext { and } 411 & 	ext { b) } 77 	ext { and } 242 & 	ext { c) } 56 	ext { and } 63\end{array} $$

UpStudy AI · Accepted Answer

To find a common prime factor for each pair of numbers, we'll factorize each number into its prime components and identify the shared prime.

### a) 21 and 411

**Factorization:**
- **21:** $$ 21 = 3 	imes 7 $$
- **411:** 
  - Sum of digits: $$ 4 + 1 + 1 = 6 $$ (divisible by 3)
  - $$ 411 \div 3 = 137 $$
  - $$ 137 $$ is a prime number.
  - So, $$ 411 = 3 	imes 137 $$

**Common Prime Factor:** **3**

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### b) 77 and 242

**Factorization:**
- **77:** $$ 77 = 7 	imes 11 $$
- **242:**
  - $$ 242 \div 2 = 121 $$
  - $$ 121 = 11 	imes 11 $$
  - So, $$ 242 = 2 	imes 11^2 $$

**Common Prime Factor:** **11**

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### c) 56 and 63

**Factorization:**
- **56:** 
  - $$ 56 = 7 	imes 8 $$
  - $$ 8 = 2^3 $$
  - So, $$ 56 = 7 	imes 2^3 $$
- **63:**
  - $$ 63 = 7 	imes 9 $$
  - $$ 9 = 3^2 $$
  - So, $$ 63 = 7 	imes 3^2 $$

**Common Prime Factor:** **7**

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**Summary of Common Prime Factors:**
- **a)** 3
- **b)** 11
- **c)** 7
- **a)** 3 - **b)** 11 - **c)** 7

Find a common prime factor in each of these pairs of numbers.

Upstudy AI Solution

Answer

Solution

a) 21 and 411

b) 77 and 242

c) 56 and 63

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