`N=70!xx69!xx68!xx….3!xx2!xx1!`
Which of the following represents the 147th digit from the right end of N?

Options

A. 2
B. 0
C. 5
D. 7

Answer

To solve the problem of finding the 147th digit from the right end of the number \( N = 70! \times 69! \times 68! \times \ldots \times 3! \times 2! \times 1! \), we need to determine how many trailing zeros are present in \( N \).

### Step-by-Step Solution:

1. **Understanding Trailing Zeros**:
Trailing zeros in a factorial are produced by the factors of 10 in the number, which are formed by pairs of 2 and 5. Since there are generally more factors of 2 than 5, the number of trailing zeros is determined by the number of times 5 is a factor in the factorial.

2. **Calculating Trailing Zeros for Each Factorial**:
We need to calculate the number of trailing zeros contributed by each factorial from \( 1! \) to \( 70! \).

The formula to find the number of trailing zeros in \( n! \) is:
\[ \text{Trailing Zeros}(n!) = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \ldots \]

3. **Calculating Trailing Zeros for Each Factorial**:
- For \( 1! \) to \( 4! \):
\[ \text{Trailing Zeros} = 0 \]
- For \( 5! \) to \( 9! \):
\[ \text{Trailing Zeros} = 1 \quad (\text{from } 5!) \]
- For \( 10! \) to \( 14! \):
\[ \text{Trailing Zeros} = 2 \quad (\text{from } 10!) \]
- For \( 15! \) to \( 19! \):
\[ \text{Trailing Zeros} = 3 \quad (\text{from } 15!) \]
- Continuing this way, we can calculate the trailing zeros for all factorials up to \( 70! \).

4. **Summing Up the Trailing Zeros**:
By calculating the trailing zeros for each factorial up to \( 70! \), we find:
- \( 70! \): 14 zeros
- \( 69! \): 13 zeros
- \( 68! \): 13 zeros
- ...
- \( 5! \): 1 zero
- \( 4! \): 0 zeros
- \( 3! \): 0 zeros
- \( 2! \): 0 zeros
- \( 1! \): 0 zeros

When we sum all these zeros, we find that there are a total of **155 trailing zeros** in \( N \).

5. **Finding the 147th Digit from the Right**:
Since there are 155 trailing zeros in \( N \), the 147th digit from the right end of \( N \) must also be a zero.

### Conclusion:
The 147th digit from the right end of \( N \) is **0**.