IF a is a factor of 32 and b is a factor of 45 , the product of a and b could NOT be which of the following ?

Options

A. `1,440`
B. `288`
C. `80`
D. `54`

Answer

To solve the problem, we need to determine the factors of 32 and 45, then find the products of these factors to see which option cannot be formed.

### Step-by-Step Solution:

1. **Find the Factors of 32:**
- The factors of 32 are: 1, 2, 4, 8, 16, and 32.

2. **Find the Factors of 45:**
- The factors of 45 are: 1, 3, 5, 9, 15, and 45.

3. **Calculate Possible Products of a and b:**
- We will multiply each factor of 32 with each factor of 45 and list the products.
- For example:
- \(1 \times 1 = 1\)
- \(1 \times 3 = 3\)
- \(1 \times 5 = 5\)
- \(1 \times 9 = 9\)
- \(1 \times 15 = 15\)
- \(1 \times 45 = 45\)
- \(2 \times 1 = 2\)
- \(2 \times 3 = 6\)
- \(2 \times 5 = 10\)
- \(2 \times 9 = 18\)
- \(2 \times 15 = 30\)
- \(2 \times 45 = 90\)
- \(4 \times 1 = 4\)
- \(4 \times 3 = 12\)
- \(4 \times 5 = 20\)
- \(4 \times 9 = 36\)
- \(4 \times 15 = 60\)
- \(4 \times 45 = 180\)
- \(8 \times 1 = 8\)
- \(8 \times 3 = 24\)
- \(8 \times 5 = 40\)
- \(8 \times 9 = 72\)
- \(8 \times 15 = 120\)
- \(8 \times 45 = 360\)
- \(16 \times 1 = 16\)
- \(16 \times 3 = 48\)
- \(16 \times 5 = 80\)
- \(16 \times 9 = 144\)
- \(16 \times 15 = 240\)
- \(16 \times 45 = 720\)
- \(32 \times 1 = 32\)
- \(32 \times 3 = 96\)
- \(32 \times 5 = 160\)
- \(32 \times 9 = 288\)
- \(32 \times 15 = 480\)
- \(32 \times 45 = 1440\)

4. **List All Unique Products:**
- The unique products obtained from the above calculations are:
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 120, 144, 160, 180, 240, 288, 360, 480, 720, 1440.

5. **Identify the Options:**
- The options provided are:
A) 1440
B) 288
C) 80
D) 54

6. **Determine Which Product Cannot Be Formed:**
- From the list of unique products, we see that 54 is present. However, it cannot be formed by multiplying any factor of 32 with any factor of 45.
- Therefore, the product of a and b could NOT be **54**.

### Final Answer:
The product of a and b could NOT be **54**.