What is the remainder when the product `1991xx1992xx2000` is divided by 7 ?

Answer

To find the remainder when the product \( 1991 \times 1992 \times 2000 \) is divided by 7, we can follow these steps:

### Step 1: Find the remainders of each number when divided by 7
1. **Calculate \( 1991 \mod 7 \)**:
- Divide \( 1991 \) by \( 7 \):
- \( 1991 \div 7 = 284 \) remainder \( 3 \) (since \( 7 \times 284 = 1988 \) and \( 1991 - 1988 = 3 \))
- So, \( 1991 \equiv 3 \mod 7 \).

2. **Calculate \( 1992 \mod 7 \)**:
- Since \( 1992 = 1991 + 1 \), we can find:
- \( 1992 \equiv 3 + 1 \equiv 4 \mod 7 \).

3. **Calculate \( 2000 \mod 7 \)**:
- Divide \( 2000 \) by \( 7 \):
- \( 2000 \div 7 = 285 \) remainder \( 5 \) (since \( 7 \times 285 = 1995 \) and \( 2000 - 1995 = 5 \))
- So, \( 2000 \equiv 5 \mod 7 \).

### Step 2: Multiply the remainders
Now we multiply the remainders we found:
\[ (1991 \mod 7) \times (1992 \mod 7) \times (2000 \mod 7) \equiv 3 \times 4 \times 5 \mod 7 \]

### Step 3: Calculate \( 3 \times 4 \times 5 \)
1. First, calculate \( 3 \times 4 = 12 \).
2. Next, multiply \( 12 \times 5 = 60 \).

### Step 4: Find the remainder of \( 60 \) when divided by \( 7 \)
Now we need to find \( 60 \mod 7 \):
- Divide \( 60 \) by \( 7 \):
- \( 60 \div 7 = 8 \) remainder \( 4 \) (since \( 7 \times 8 = 56 \) and \( 60 - 56 = 4 \)).
- So, \( 60 \equiv 4 \mod 7 \).

### Final Answer
Thus, the remainder when \( 1991 \times 1992 \times 2000 \) is divided by \( 7 \) is \( \boxed{4} \).