Find the number of maximum possible HCF values for N, N + 240?

Options

A. 12
B. 13
C. 20
D. 18

Answer

To find the number of maximum possible HCF values for \( N \) and \( N + 240 \), we can follow these steps:

### Step 1: Understand the relationship between the numbers
We need to find the HCF (Highest Common Factor) of two numbers: \( N \) and \( N + 240 \). The HCF of these two numbers can be expressed in terms of their difference.

**Hint:** The HCF of two numbers \( a \) and \( b \) can also be found using their difference \( |a - b| \).

### Step 2: Express the HCF in terms of the difference
The difference between the two numbers is:
\[ (N + 240) - N = 240 \]
Thus, we can say:
\[ \text{HCF}(N, N + 240) = \text{HCF}(N, 240) \]

**Hint:** The HCF remains the same when you replace one number with the difference of the two.

### Step 3: Find the factors of 240
Next, we need to find the factors of 240 to determine the possible values of HCF. First, we can find the prime factorization of 240:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

**Hint:** Prime factorization helps in identifying all possible factors of a number.

### Step 4: Calculate the number of factors
To find the total number of factors, we use the formula:
\[ \text{Number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \( e_1, e_2, e_3, \ldots \) are the powers of the prime factors.

For 240:
- The power of 2 is 4, so \( e_1 = 4 \)
- The power of 3 is 1, so \( e_2 = 1 \)
- The power of 5 is 1, so \( e_3 = 1 \)

Thus, the number of factors is:
\[ (4 + 1)(1 + 1)(1 + 1) = 5 \times 2 \times 2 = 20 \]

**Hint:** Each exponent in the prime factorization contributes to the total number of factors.

### Step 5: Conclusion
Therefore, the maximum possible HCF values for \( N \) and \( N + 240 \) is 20.

**Final Answer:** There are 20 different possible values for the HCF of \( N \) and \( N + 240 \).

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