If the four digit number 27N4 is divisible by 11, what is the value of the digit N?

Introduction / Context: This question checks the divisibility rule for 11 applied to a four digit number with an unknown digit. It is a common type of problem in number system sections of aptitude tests.

Given Data / Assumptions: - The number is 27N4, where N is a single digit from 0 to 9. - The number 27N4 is divisible by 11. - We are required to find the digit N.

Concept / Approach: The divisibility rule for 11 states that a number is divisible by 11 if and only if the alternating sum of its digits is a multiple of 11 (including 0). For a four digit number abcd, we compute a − b + c − d and check whether the result is 0, ±11, ±22, and so on. We apply this rule to 27N4.

Step-by-Step Solution: Write the digits: a = 2, b = 7, c = N, d = 4. Compute the alternating sum: a − b + c − d = 2 − 7 + N − 4. Simplify: 2 − 7 = −5, and −5 − 4 = −9, so the expression becomes N − 9. For divisibility by 11, N − 9 must be equal to 0, 11, −11, and so on. The only digit N between 0 and 9 that satisfies this is N = 9, because 9 − 9 = 0.

Verification / Alternative Check: Substitute N = 9 to form the number 2794 and compute the alternating sum: 2 − 7 + 9 − 4 = 0. Since 0 is a multiple of 11, 2794 is divisible by 11 according to the rule. None of the other digits would give an alternating sum that is a multiple of 11 while remaining a valid single digit.

Why Other Options Are Wrong: If N = 2, the alternating sum would be 2 − 7 + 2 − 4 = −7, not a multiple of 11. If N = 7, the sum becomes 2 − 7 + 7 − 4 = −2, again not a multiple of 11. If N = 6, the sum is 2 − 7 + 6 − 4 = −3, also invalid. If N = 1, the sum is 2 − 7 + 1 − 4 = −8, which does not satisfy the rule.

Common Pitfalls: Learners sometimes misapply the rule by summing in the wrong pattern or by using absolute values instead of an alternating sum. Others forget that the result can be 0 as well as ±11, ±22, and so on. Carefully following the pattern a − b + c − d and checking whether the result is 0 or a multiple of 11 avoids such errors.

Final Answer: The value of N that makes 27N4 divisible by 11 is 9.