Assertion(A): The HCF of 131 and 201=1
Reason (R):The HCF of co prime numbers is 1.

Options

A. Both A and R are true and R is the correct explanation for A .
B. Both A and Rare true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the problem, we need to determine the truth of the assertion (A) and the reason (R) given in the question.

**Step 1: Understanding the Assertion (A)**
The assertion states that the HCF (Highest Common Factor) of 131 and 201 is 1.

**Step 2: Finding the HCF of 131 and 201**
To find the HCF, we can use the prime factorization method.

- **Prime Factorization of 131:**
131 is a prime number, so its only factors are 1 and 131 itself.
Thus, the prime factorization of 131 is:
\( 131 = 1 \times 131 \)

- **Prime Factorization of 201:**
To factor 201, we can divide it by its smallest prime factor, which is 3.
\( 201 \div 3 = 67 \)
Now, 67 is also a prime number.
Thus, the prime factorization of 201 is:
\( 201 = 3 \times 67 \)

**Step 3: Identifying Common Factors**
Now, we need to identify the common factors of both numbers.
- The factors of 131 are: 1, 131
- The factors of 201 are: 1, 3, 67, 201

The only common factor between 131 and 201 is 1.

**Step 4: Conclusion for Assertion (A)**
Since the only common factor is 1, we conclude that the HCF of 131 and 201 is indeed 1. Therefore, assertion (A) is true.

**Step 5: Understanding the Reason (R)**
The reason states that the HCF of co-prime numbers is 1. By definition, co-prime numbers are numbers that have no common factors other than 1.

**Step 6: Validating the Reason (R)**
Since 131 and 201 do not share any prime factors (they are co-prime), the reason (R) is also true.

**Step 7: Final Conclusion**
Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion. Therefore, the correct answer is that both statements are true, and the reason is a correct explanation of the assertion.