Find the no. of zeros in expression :
`(8^(253) - 8^(252) - 8^(251))(3^(221) - 3^(220) - 3^(219))`

Options

A. 1
B. 2
C. 3
D. 0

Answer

To find the number of zeros in the expression \( (8^{253} - 8^{252} - 8^{251})(3^{221} - 3^{220} - 3^{219}) \), we need to analyze each part of the expression separately and then combine the results.

### Step 1: Simplify the first part \( 8^{253} - 8^{252} - 8^{251} \)

1. **Factor out the common term**:
\[ 8^{251}(8^2 - 8 - 1) \]

2. **Calculate \( 8^2 - 8 - 1 \)**:
\[ 8^2 = 64 \quad \Rightarrow \quad 64 - 8 - 1 = 55 \]

3. **So, we have**:
\[ 8^{253} - 8^{252} - 8^{251} = 8^{251} \times 55 \]

### Step 2: Simplify the second part \( 3^{221} - 3^{220} - 3^{219} \)

1. **Factor out the common term**:
\[ 3^{219}(3^2 - 3 - 1) \]

2. **Calculate \( 3^2 - 3 - 1 \)**:
\[ 3^2 = 9 \quad \Rightarrow \quad 9 - 3 - 1 = 5 \]

3. **So, we have**:
\[ 3^{221} - 3^{220} - 3^{219} = 3^{219} \times 5 \]

### Step 3: Combine the results

Now we can combine both parts:
\[ (8^{251} \times 55)(3^{219} \times 5) = 8^{251} \times 3^{219} \times 55 \times 5 \]

### Step 4: Count the factors of 2 and 5

1. **Calculate the factors of 2**:
- \( 8^{251} = (2^3)^{251} = 2^{753} \)
- \( 55 \) has no factors of 2.
- \( 5 \) has no factors of 2.
- Total factors of 2 = \( 753 \)

2. **Calculate the factors of 5**:
- \( 55 = 5 \times 11 \) contributes 1 factor of 5.
- \( 5 \) contributes 1 factor of 5.
- Total factors of 5 = \( 1 + 1 = 2 \)

### Step 5: Determine the number of zeros

The number of zeros at the end of the number is determined by the minimum of the number of pairs of factors of 2 and 5:
\[ \text{Number of zeros} = \min(753, 2) = 2 \]

Thus, the final answer is:
\[ \text{Number of zeros} = 2 \]