Find the greatest number of five digits which when divided by 3, 5, 8, 12 have 2 as remainder :

Options

A. 99999
B. 99958
C. 99960
D. 99962

Answer

To solve the problem of finding the greatest five-digit number that leaves a remainder of 2 when divided by 3, 5, 8, and 12, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the greatest five-digit number**:
The greatest five-digit number is 99999.

**Hint**: Remember that five-digit numbers range from 10000 to 99999.

2. **Determine the required condition**:
We need a number that, when divided by 3, 5, 8, and 12, leaves a remainder of 2. This means that if we subtract 2 from our number, it should be divisible by the least common multiple (LCM) of these divisors.

**Hint**: Understand that if a number leaves a remainder of 2 when divided by another number, subtracting that remainder gives a number that is divisible by that other number.

3. **Calculate the LCM of 3, 5, 8, and 12**:
- The prime factorization of the numbers is:
- 3 = 3
- 5 = 5
- 8 = 2^3
- 12 = 2^2 * 3
- The LCM is found by taking the highest power of each prime:
- LCM = 2^3 * 3^1 * 5^1 = 120

**Hint**: To find the LCM, consider the highest power of each prime factor present in the numbers.

4. **Adjust the greatest five-digit number**:
We need to find the largest number less than or equal to 99999 that is divisible by 120. To do this, we can divide 99999 by 120 and find the largest integer.

- 99999 ÷ 120 = 833.325
- The largest integer is 833.

**Hint**: When dividing, always round down to the nearest whole number to find the largest integer.

5. **Calculate the largest multiple of 120**:
Multiply the integer found in the previous step by 120 to find the largest multiple of 120 that is less than or equal to 99999.

- 833 * 120 = 99960

**Hint**: Multiplying the integer found by the LCM gives you the largest number that meets the divisibility condition.

6. **Add the remainder back**:
Since we need a number that leaves a remainder of 2, we add 2 to the result from the previous step:

- 99960 + 2 = 99962

**Hint**: Always remember to add back the remainder after finding the largest multiple.

### Final Answer:
The greatest five-digit number which, when divided by 3, 5, 8, and 12, has a remainder of 2 is **99962**.