Find the least positive integer N such that: • When N is divided by 5, 6, 7, and 8, it leaves a remainder of 3 each time, and • When N is divided by 9, it leaves no remainder. What is the value of N?

Introduction / Context:
This problem involves modular arithmetic and least common multiples. When a number leaves the same remainder upon division by several divisors, we can convert that statement into one involving the LCM of those divisors. The added condition that the number is divisible by 9 allows us to select the correct multiple.

Given Data / Assumptions:

• N mod 5 = 3
• N mod 6 = 3
• N mod 7 = 3
• N mod 8 = 3
• N mod 9 = 0

Concept / Approach:
If N leaves remainder 3 when divided by 5, 6, 7, and 8, then N - 3 is divisible by all of these numbers. Therefore:
N - 3 is a multiple of LCM(5, 6, 7, 8)
So we can write:
N = 840k + 3
for some positive integer k, where 840 is the LCM. Then we use the condition N mod 9 = 0 to determine the smallest suitable k.

Step-by-Step Solution:
Step 1: Compute LCM(5, 6, 7, 8).5 = 5, 6 = 2 * 3, 7 = 7, 8 = 2^3.LCM = 2^3 * 3 * 5 * 7 = 8 * 3 * 5 * 7 = 840.Step 2: Write N as N = 840k + 3.Step 3: Apply divisibility by 9: 840k + 3 ≡ 0 (mod 9).840 mod 9 = 3, so 840k + 3 ≡ 3k + 3 (mod 9).Thus 3k + 3 ≡ 0 (mod 9) ⇒ 3(k + 1) ≡ 0 (mod 9).Divide by 3: k + 1 ≡ 0 (mod 3) ⇒ k ≡ 2 (mod 3).Step 4: The smallest positive k that satisfies this is k = 2.Step 5: Therefore N = 840 * 2 + 3 = 1683.

Verification / Alternative check:
Check N = 1683: 1683 - 3 = 1680, which is divisible by 5, 6, 7, and 8. Also, 1 + 6 + 8 + 3 = 18, which is divisible by 9, so 1683 is divisible by 9. No smaller positive k gives a valid N, so 1683 is the least such number.

Why Other Options Are Wrong:
1677: 1677 - 3 = 1674 is not divisible by 8.2523 and 3363: They can be written in the form 840k + 3, but correspond to larger k values, so they are not the least.8403: Much larger and not the minimum solution.

Common Pitfalls:
Forgetting to transform N to N - 3 before applying the LCM idea.Confusing LCM and HCF when dealing with common divisibility.Not enforcing the condition that N must be the smallest positive integer that works.

Final Answer:
1683