Final answer:
The greatest 5-digit number divisible by 9, 12, 24, and 5 with remainders of 3, 6, 18, and 38, respectively, is found by subtracting multiples of the LCM of 9, 24, and 5 (which is 360) from the greatest 5-digit number until the conditions are met, which gives 99963.
Step-by-step explanation:
To find the greatest 5-digit number that leaves specific remainders when divided by 9, 12, 24, and 5, we need to understand the concept of Least Common Multiple (LCM) and Chinese Remainder Theorem. Since 24 is a multiple of both 12 and 8, we initially focus on the LCM of 9, 24, and 5, which is 360. The number must be 3 more than a multiple of 360 to satisfy the remainders of 3 when divided by 9, 18 when divided by 24, and 38 when divided by 5.
By trial and error, starting from the greatest 5-digit number (99999) and subtracting 360 successively until we get a number that leaves a remainder of 3 when divided by 9, a remainder of 18 when divided by 24, and a remainder of 38 when divided by 5. We find that the number is 99963. So the correct answer is option C) 99963.