Find the greatest number of five digits which when divided by 4,6,14 and 20 leaves respectively 1,3, 11 and 17 as remainders

Options

A. 99930
B. 99960
C. 99997
D. 99957

Answer

To find the greatest five-digit number that leaves specific remainders when divided by 4, 6, 14, and 20, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Remainders and Divisors**:
- When divided by 4, remainder is 1.
- When divided by 6, remainder is 3.
- When divided by 14, remainder is 11.
- When divided by 20, remainder is 17.

2. **Calculate the LCM of the Divisors**:
- The divisors are 4, 6, 14, and 20.
- To find the LCM, we can factor each number:
- 4 = 2²
- 6 = 2 × 3
- 14 = 2 × 7
- 20 = 2² × 5
- The LCM is found by taking the highest power of each prime factor:
- LCM = 2² × 3 × 5 × 7 = 420.

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

4. **Determine the Remainder When Dividing by LCM**:
- Divide 99999 by 420 to find the remainder:
- 99999 ÷ 420 = 238 (approximately).
- 238 × 420 = 99960.
- Remainder = 99999 - 99960 = 39.

5. **Adjust the Remainder**:
- According to the problem, we need the number to leave specific remainders:
- For 4: 1 (so we need 0 when subtracting from 1)
- For 6: 3 (so we need 3 when subtracting from 3)
- For 14: 11 (so we need 3 when subtracting from 11)
- For 20: 17 (so we need 3 when subtracting from 17)
- The common difference is 3 (since all the differences are 3).

6. **Calculate the Required Number**:
- Since the remainder we have is 39 and we need to adjust it down by 3, we subtract:
- Required number = 99999 - 39 + 3 = 99963.

7. **Final Answer**:
- The greatest five-digit number that meets the criteria is **99963**.