HCF of 315, 441, 357 is

Question

Accepted Answer

To determine the Highest Common Factor (HCF) of the numbers 315, 441, and 357, we need to find the largest number that divides each of these numbers without leaving a remainder. The correct answer is Option 3, which is 21. Let's go through the explanation step by step:

1. **Prime Factorization:**
   - **315:** The prime factorization of 315 is $3^2 \times 5 \times 7$.
   - **441:** The prime factorization of 441 is $3^2 \times 7^2$.
   - **357:** The prime factorization of 357 is $3 \times 7 \times 17$.

2. **Common Factors:**
   - From the prime factorizations, we identify the common prime factors among 315, 441, and 357.
   - The common prime factors are 3 and 7.

3. **Lowest Powers of Common Factors:**
   - For the prime number 3, the lowest power present in all three numbers is $3^1$.
   - For the prime number 7, the lowest power present in all three numbers is $7^1$.

4. **Calculate the HCF:**
   - Multiply the lowest powers of all common prime factors: $3^1 \times 7^1 = 3 \times 7 = 21$.

Therefore, the HCF of 315, 441, and 357 is 21, which confirms that Option 3 is correct.

**Explanation for Incorrect Options:**

- **Option 1 (17):** The number 17 is not a factor of any of the given numbers (315, 441, 357) based on their prime factorizations. Therefore, it cannot be the HCF.

- **Option 2 (19):** Similarly, 19 is not a factor of any of the numbers. It does not appear in the prime factorization of 315, 441, or 357, so it cannot be the HCF.

- **Option 4 (16):** The number 16 is not a factor of any of the numbers. It is not even a prime factor of any of the numbers, as it is composed of the prime number 2, which does not appear in any of the factorizations.

In conclusion, the correct answer is Option 3, as 21 is the largest number that divides 315, 441, and 357 without leaving a remainder.