Find the last two digit in the product of -
`39 xx 55 xx 57 xx 24 xx 13872 xx 9871 `

Options

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

Answer

To find the last two digits of the product \(39 \times 55 \times 57 \times 24 \times 13872 \times 9871\), we can simplify our calculations by focusing on the product modulo 100. This allows us to find the last two digits without calculating the entire product.

### Step-by-Step Solution:

1. **Identify the Product**: We need to calculate \(P = 39 \times 55 \times 57 \times 24 \times 13872 \times 9871\).

2. **Calculate Each Factor Modulo 100**:
- \(39 \mod 100 = 39\)
- \(55 \mod 100 = 55\)
- \(57 \mod 100 = 57\)
- \(24 \mod 100 = 24\)
- \(13872 \mod 100 = 72\) (since \(13872 - 13800 = 72\))
- \(9871 \mod 100 = 71\) (since \(9871 - 9800 = 71\))

3. **Multiply the Factors Together Modulo 100**:
- Start with \(39 \times 55\):
\[ 39 \times 55 = 2145 \quad \Rightarrow \quad 2145 \mod 100 = 45 \]
- Now multiply by \(57\):
\[ 45 \times 57 = 2565 \quad \Rightarrow \quad 2565 \mod 100 = 65 \]
- Next multiply by \(24\):
\[ 65 \times 24 = 1560 \quad \Rightarrow \quad 1560 \mod 100 = 60 \]
- Now multiply by \(72\):
\[ 60 \times 72 = 4320 \quad \Rightarrow \quad 4320 \mod 100 = 20 \]
- Finally, multiply by \(71\):
\[ 20 \times 71 = 1420 \quad \Rightarrow \quad 1420 \mod 100 = 20 \]

4. **Conclusion**: The last two digits of the product \(39 \times 55 \times 57 \times 24 \times 13872 \times 9871\) are **20**.