Find the greatest number of five digits which become exactly divisible by 10, 12, 15 and 18 when 3769 is added to it

Options

A. 99819
B. 99911
C. 99900
D. 99111

Answer

To solve the problem of finding the greatest five-digit number that becomes exactly divisible by 10, 12, 15, and 18 when 3769 is added to it, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Numbers**: We need to find a five-digit number \( N \) such that \( N + 3769 \) is divisible by 10, 12, 15, and 18.

2. **Find the LCM**: To ensure that \( N + 3769 \) is divisible by all four numbers, we first need to find the least common multiple (LCM) of these numbers.
- Factor the numbers:
- \( 10 = 2 \times 5 \)
- \( 12 = 2^2 \times 3 \)
- \( 15 = 3 \times 5 \)
- \( 18 = 2 \times 3^2 \)
- The LCM is found by taking the highest power of each prime factor:
- \( 2^2 \) from 12
- \( 3^2 \) from 18
- \( 5^1 \) from 10 or 15
- Therefore, the LCM is:
\[ \text{LCM} = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 \]

3. **Find the Greatest Five-Digit Number**: The greatest five-digit number is 99999.

4. **Add 3769**: We need to check \( 99999 + 3769 \):
\[ 99999 + 3769 = 103768 \]

5. **Check Divisibility**: Now we need to find the remainder when 103768 is divided by 180:
- Perform the division:
\[ 103768 \div 180 \approx 576.5 \quad (\text{take the integer part, } 576) \]
- Calculate \( 180 \times 576 = 103680 \).
- Subtract this from 103768 to find the remainder:
\[ 103768 - 103680 = 88 \]

6. **Adjust the Number**: Since the remainder is 88, we need to subtract this from 99999 to find the required number:
\[ 99999 - 88 = 99911 \]

7. **Final Result**: The greatest five-digit number that satisfies the condition is:
\[ \boxed{99911} \]