To find the least number that satisfies the given conditions, we can use the concept of the least common multiple (LCM). The LCM of a set of numbers is the smallest multiple that is divisible by all the numbers in the set.

Let's break down the problem step by step:

Step 1: Find the LCM of 5, 6, 7, and 8
The LCM of 5, 6, 7, and 8 can be found by listing their multiples and finding the smallest common multiple. However, there is a faster way to calculate the LCM using prime factorization.

Prime factorization of each number:
- 5 = 5
- 6 = 2 * 3
- 7 = 7
- 8 = 2^3

Now, we take the highest power of each prime factor:
- 2^3 * 3 * 5 * 7 = 840

Therefore, the LCM of 5, 6, 7, and 8 is 840.

Step 2: Find the number which leaves a remainder of 3 when divided by 5, 6, 7, and 8
To find this number, we need to find a multiple of 840 that leaves a remainder of 3 when divided by 5, 6, 7, and 8.

Since 840 is the LCM of these numbers, any multiple of 840 will also be divisible by all of them. So, we need to find the smallest multiple of 840 that leaves a remainder of 3 when divided by 5, 6, 7, and 8.

We can start by adding 3 to 840 and check if it satisfies the conditions:
- 843 ÷ 5 = 168 remainder 3
- 843 ÷ 6 = 140 remainder 3
- 843 ÷ 7 = 120 remainder 3
- 843 ÷ 8 = 105 remainder 3
- 843 ÷ 9 = 93 with no remainder

Therefore, the least number that satisfies the given conditions is 843.

Step 3: Check the answer options
Now, let's check the answer options given:
- Option A: 1677 ÷ 9 = 186 with no remainder (not the correct answer)
- Option B: 1683 ÷ 9 = 187 with no remainder (correct answer)
- Option C: 2523 ÷ 9 = 280 with no remainder (not the correct answer)
- Option D: 3363 ÷ 9 = 374 with no remainder (not the correct answer)

The only option that satisfies the condition of leaving no remainder when divided by 9 is option B, which is 1683.

Therefore, the correct answer is option B.