If `a + b = n , n in I` and `ab = 60` then n cannont be equal to :

Options

A. 33
B. `-19`
C. `17`
D. `32`

Answer

To solve the problem, we need to find the values of \( n \) given that \( a + b = n \) and \( ab = 60 \). We will explore the pairs of integers \( (a, b) \) that satisfy the equation \( ab = 60 \) and then calculate \( n \) for each pair.

### Step-by-Step Solution:

1. **Identify the factors of 60**:
We start by listing the pairs of integers \( (a, b) \) such that their product is 60. The factor pairs of 60 are:
- \( (1, 60) \)
- \( (2, 30) \)
- \( (3, 20) \)
- \( (4, 15) \)
- \( (5, 12) \)
- \( (6, 10) \)

2. **Calculate \( n \) for each factor pair**:
For each pair \( (a, b) \), we calculate \( n = a + b \):
- For \( (1, 60) \): \( n = 1 + 60 = 61 \)
- For \( (2, 30) \): \( n = 2 + 30 = 32 \)
- For \( (3, 20) \): \( n = 3 + 20 = 23 \)
- For \( (4, 15) \): \( n = 4 + 15 = 19 \)
- For \( (5, 12) \): \( n = 5 + 12 = 17 \)
- For \( (6, 10) \): \( n = 6 + 10 = 16 \)

3. **Consider negative pairs**:
Since \( ab = 60 \) can also be satisfied with negative integers, we also consider the negative pairs:
- \( (-1, -60) \): \( n = -1 - 60 = -61 \)
- \( (-2, -30) \): \( n = -2 - 30 = -32 \)
- \( (-3, -20) \): \( n = -3 - 20 = -23 \)
- \( (-4, -15) \): \( n = -4 - 15 = -19 \)
- \( (-5, -12) \): \( n = -5 - 12 = -17 \)
- \( (-6, -10) \): \( n = -6 - 10 = -16 \)

4. **Compile the possible values of \( n \)**:
From the calculations, the possible values of \( n \) are:
- Positive values: \( 61, 32, 23, 19, 17, 16 \)
- Negative values: \( -61, -32, -23, -19, -17, -16 \)

5. **Determine which value of \( n \) cannot be achieved**:
The problem asks which value \( n \) cannot be equal to. From our calculations, we see that all integers except for \( 33 \) can be achieved as \( n \). Therefore, \( n \) cannot be equal to \( 33 \).

### Final Answer:
Thus, \( n \) cannot be equal to **33**.