N is the largest two digit number that leaves remainders 1, 2 and 4 when divided by 3, 4 and 6 respectively. What is the remainder when N is divided by 5?

Introduction / Context:
This question tests your understanding of simultaneous congruences and modular arithmetic. You are given conditions on the remainders when a two digit number N is divided by 3, 4, and 6, and you must find the largest two digit number that satisfies all these conditions. After that, you must compute the remainder when that number is divided by 5. This is a classic number system problem that often appears in quantitative aptitude exams.

Given Data / Assumptions:

• N is a two digit integer, so 10 ≤ N ≤ 99.
• N leaves a remainder of 1 when divided by 3.
• N leaves a remainder of 2 when divided by 4.
• N leaves a remainder of 4 when divided by 6.
• We must find the largest such N and then determine N mod 5.

Concept / Approach:
The conditions can be written using congruence notation as follows: N ≡ 1 (mod 3), N ≡ 2 (mod 4), and N ≡ 4 (mod 6). We can search systematically through the two digit integers that satisfy these conditions, or we can use the pattern that such numbers repeat modulo the least common multiple of 3, 4, and 6. Once we find all candidate values, we select the largest one under 100 and then compute its remainder when divided by 5.

Step-by-Step Solution:
Step 1: Express the conditions as congruences: N ≡ 1 (mod 3), N ≡ 2 (mod 4), N ≡ 4 (mod 6). Step 2: The least common multiple of 3, 4, and 6 is 12. Step 3: We can search for numbers between 10 and 99 that satisfy all three congruences. The valid solutions form a repeating pattern with period 12. Step 4: By checking or using a prepared list, we find that the numbers 10, 22, 34, 46, 58, 70, 82, and 94 satisfy all three remainder conditions. Step 5: Among these, the largest two digit number is N = 94. Step 6: Now we must calculate the remainder when 94 is divided by 5. Step 7: Divide 94 by 5. We have 5 * 18 = 90, and 5 * 19 = 95 which is too large, so the quotient is 18 and the remainder is 94 − 90 = 4. Step 8: Therefore, the remainder when N is divided by 5 is 4.

Verification / Alternative check:
Verify the congruence conditions for N = 94. For division by 3, 94 divided by 3 gives 31 with remainder 1, so N ≡ 1 (mod 3). For division by 4, 94 divided by 4 gives 23 with remainder 2, so N ≡ 2 (mod 4). For division by 6, 94 divided by 6 gives 15 with remainder 4, so N ≡ 4 (mod 6). These checks confirm that 94 is valid. Then 94 divided by 5 gives remainder 4, confirming the final answer.

Why Other Options Are Wrong:
Remainders 2, 0, 1, or 3 would come from different two digit values that either do not satisfy the three simultaneous congruences or are not the largest valid N. For example, a value like 70 also satisfies the congruences but 70 mod 5 equals 0. However, 70 is not the largest valid two digit number; 94 exceeds it and also meets the conditions. Therefore the only correct remainder for the largest allowed N is 4.

Common Pitfalls:
Students often mix up the conditions or fail to check all of them together. Checking only one or two congruences can lead to selecting an incorrect N. Another common mistake is to forget the requirement that N must be the largest two digit number satisfying the conditions, which can lead to choosing a smaller valid number. Careful use of modular arithmetic and systematic checking of candidates avoids these issues.

Final Answer:
The remainder when N is divided by 5 is 4.