Find the greatest number of 4 digits and the least number of 5 digits which when divided by 789 leave a remainder 5 in each case.

Options

A. 9473, 10262
B. 9573, 10362
C. 967310462
D. 9676, 10465

Answer

To solve the problem of finding the greatest 4-digit number and the least 5-digit number that, when divided by 789, leave a remainder of 5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Greatest 4-Digit Number:**
The greatest 4-digit number is 9999.

2. **Calculate the Remainder of 9999 when Divided by 789:**
We need to divide 9999 by 789 to find the remainder.
\[ 9999 \div 789 \approx 12.66 \quad (\text{Take the integer part, which is } 12) \]
Now, calculate \(12 \times 789\):
\[ 12 \times 789 = 9468 \]
Now, subtract this from 9999 to find the remainder:
\[ 9999 - 9468 = 531 \]

3. **Adjust to Get the Desired Remainder of 5:**
Since we need a remainder of 5, we will subtract the excess from the greatest 4-digit number:
\[ 9999 - 531 + 5 = 9999 - 526 = 9473 \]
Thus, the greatest 4-digit number that leaves a remainder of 5 when divided by 789 is **9473**.

4. **Identify the Least 5-Digit Number:**
The least 5-digit number is 10000.

5. **Calculate the Remainder of 10000 when Divided by 789:**
We need to divide 10000 by 789.
\[ 10000 \div 789 \approx 12.67 \quad (\text{Take the integer part, which is } 12) \]
Now, calculate \(12 \times 789\):
\[ 12 \times 789 = 9468 \]
Now, subtract this from 10000 to find the remainder:
\[ 10000 - 9468 = 532 \]

6. **Adjust to Get the Desired Remainder of 5:**
Since we need a remainder of 5, we will add the difference to the least 5-digit number:
\[ 10000 + (789 - 532) + 5 = 10000 + 257 + 5 = 10000 + 262 = 10262 \]
Thus, the least 5-digit number that leaves a remainder of 5 when divided by 789 is **10262**.

### Final Answers:
- The greatest 4-digit number is **9473**.
- The least 5-digit number is **10262**.