Find the greatest number which will divide the greatest 3-digit number and the greatest 4-digit number exactly

Options

A. 9
B. 8
C. 7
D. 3

Answer

To find the greatest number that will divide both the greatest 3-digit number and the greatest 4-digit number exactly, we can follow these steps:

### Step 1: Identify the greatest 3-digit and 4-digit numbers
- The greatest 3-digit number is **999**.
- The greatest 4-digit number is **9999**.

### Step 2: Find the factors of both numbers
To find the greatest number that divides both 999 and 9999, we need to find the greatest common factor (GCF) of these two numbers.

### Step 3: Prime Factorization
1. **Prime factorization of 999**:
- 999 can be divided by 3:
- 999 ÷ 3 = 333
- 333 can also be divided by 3:
- 333 ÷ 3 = 111
- 111 can also be divided by 3:
- 111 ÷ 3 = 37
- So, the prime factorization of 999 is:
- \( 999 = 3^3 \times 37 \)

2. **Prime factorization of 9999**:
- 9999 can also be divided by 3:
- 9999 ÷ 3 = 3333
- 3333 can be divided by 3:
- 3333 ÷ 3 = 1111
- 1111 can be divided by 11:
- 1111 ÷ 11 = 101
- So, the prime factorization of 9999 is:
- \( 9999 = 3^2 \times 11 \times 101 \)

### Step 4: Identify the common factors
- From the prime factorizations:
- 999: \( 3^3 \times 37 \)
- 9999: \( 3^2 \times 11 \times 101 \)

The common factor is \( 3^2 \) since it is the lowest power of 3 present in both factorizations.

### Step 5: Calculate the GCF
- The GCF is:
- \( 3^2 = 9 \)

### Conclusion
The greatest number that will divide both the greatest 3-digit number (999) and the greatest 4-digit number (9999) exactly is **9**.

---