Final answer:
The least number that meets the criteria is 1683, which is 3 more than the LCM of 5, 6, 7, and 8 and also divisible by 9.
Step-by-step explanation:
To find the least number which, when divided by 5, 6, 7, and 8 gives the remainder 3 but is divisible by 9, we need to find a number that is 3 more than the least common multiple (LCM) of 5, 6, 7, and 8 and also divisible by 9. The LCM of 5, 6, 7, and 8 is 840. Therefore, adding 3, we get 843. However, 843 is not divisible by 9. To find the next number that satisfies the given remainder condition, we add multiples of the LCM (840) to 843 until we find a number divisible by 9.
Upon trying a few multiples, we find that 843 + 840× 1 = 1683, which is divisible by 9. Therefore, 1683 is the correct answer, which is option C.