To find the greatest five-digit number which when divided by 3, 8, and 12 leaves no remainder (i.e., is divisible by these numbers), we proceed as follows:
- Identify the Greatest 5-digit Number: The greatest five-digit number is 99999.
- Calculate the Least Common Multiple (LCM) of 3, 8, and 12:
- LCM(3, 8, 12) = 24 × 5 (since 3 × 8 = 24, and the LCM with 12 is 24)
- In detail, LCM of 3, 8, and 12 is 24.
- Find the greatest multiple of 24 that is less than or equal to 99999 by dividing 99999 by 24:
- ⌊9999924⌋=4166\lfloor \frac{99999}{24}\rfloor =4166⌊2499999⌋=4166
- Then multiply back: 4166×24=999844166\times 24=999844166×24=99984
Hence, the greatest five-digit number divisible by 3, 8, and 12 is 99984. If your question is about the greatest five-digit number which, when divided by 3, 5, 8, and 12, leaves a remainder (such as 2), that number would be different, but for divisibility specifically by 3, 8, and 12, the answer is 99984.
