M is defined as the largest four digit number which, when divided by 4, 5, 6 and 7, leaves remainders 2, 3, 4 and 5 respectively. What will be the remainder when this number M is divided by 9?

Introduction / Context:
This is a number theory and modular arithmetic question involving remainders. We are told that a certain four digit number M leaves specific remainders when divided by 4, 5, 6 and 7, and we must find the remainder when M is divided by 9. Such problems are very common in aptitude tests and require understanding congruences and least common multiples, as well as some logical reasoning to identify the largest number with the stated properties.

Given Data / Assumptions:

• M is a four digit number (1000 ≤ M ≤ 9999).
• M ≡ 2 (mod 4).
• M ≡ 3 (mod 5).
• M ≡ 4 (mod 6).
• M ≡ 5 (mod 7).
• We want the remainder when M is divided by 9.

Concept / Approach:
Notice that for each divisor k in {4, 5, 6, 7}, the remainder is k - 2. This can be rewritten as: M ≡ -2 (mod 4), M ≡ -2 (mod 5), M ≡ -2 (mod 6), M ≡ -2 (mod 7). Hence M + 2 is divisible by 4, 5, 6 and 7. This means that M + 2 is a multiple of the least common multiple of 4, 5, 6 and 7. Once we find that least common multiple, we can identify the largest multiple that leads to a four digit M and then compute its remainder modulo 9.

Step-by-Step Solution:
Step 1: Compute L = lcm(4, 5, 6, 7). Prime factorizations: 4 = 2^2, 5 = 5, 6 = 2 * 3, 7 = 7. The least common multiple takes the highest powers of all primes: 2^2 * 3 * 5 * 7 = 4 * 3 * 5 * 7 = 420. Step 2: We have M + 2 = 420k for some integer k, and M is four digit, so 1000 ≤ M ≤ 9999. Thus 1002 ≤ 420k ≤ 10001. Step 3: Find the largest integer k such that 420k ≤ 10001. Compute 420 * 23 = 9660 and 420 * 24 = 10080 (too large). So the largest valid k is 23, which gives M + 2 = 9660. Step 4: Therefore, M = 9660 - 2 = 9658. Step 5: Now find the remainder when M is divided by 9. Compute 9658 ÷ 9. Since 9 * 1073 = 9657, the remainder is 1.

Verification / Alternative check:
We can directly check that 9658 satisfies the original remainder conditions. Dividing 9658 by 4 gives a remainder of 2, by 5 gives a remainder of 3, by 6 gives a remainder of 4 and by 7 gives a remainder of 5. This confirms that M was correctly found. Finally, checking modulo 9 confirms 9658 ≡ 1 (mod 9), consistent with the calculations above.

Why Other Options Are Wrong:
Option a, remainder 2, and option c, remainder 3, would require 9658 minus 2 or 3 to be divisible by 9, which is not the case. Option d, 6, and option e, 0, also do not match the digit sum check for divisibility by 9. The digit sum of 9658 is 9 + 6 + 5 + 8 = 28, and 28 leaves remainder 1 upon division by 9, which directly supports remainder 1. Thus only option b is correct.

Common Pitfalls:
A frequent mistake is to try to solve each congruence separately without recognizing the simple pattern M ≡ -2 (mod k). Another is to compute the least common multiple incorrectly, for example omitting the factor 3 from 6 or not using the highest powers. Some students also forget that M must be a four digit number and pick a smaller multiple. Finally, arithmetic slips when dividing by 9 can lead to an incorrect remainder. Using the digit sum test for divisibility by 9 is a quick and reliable way to avoid such mistakes.

Final Answer:
Therefore, when M is divided by 9, the remainder is 1.