If the four-digit number 56M4 is completely divisible by 11, then what digit does M represent in the number 56M4?

Introduction / Context:
This question tests divisibility rules, a common topic in quantitative aptitude sections of competitive exams. Instead of performing full long division, examinees are expected to recall and apply the shortcut rule for divisibility by 11. The number given is 56M4, where M is a digit from 0 to 9. The task is to determine which value of M makes the entire four digit number divisible by 11. Understanding and using divisibility rules saves time during exams and reduces calculation errors.

Given Data / Assumptions:

• We have a four digit number of the form 56M4.
• M is a single digit (0 to 9).
• The number 56M4 must be exactly divisible by 11 without any remainder.
• We use the standard divisibility rule for 11.

Concept / Approach:
The divisibility rule for 11 states that a number is divisible by 11 if and only if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is either 0 or a multiple of 11 (for example 0, 11, 22, and so on). For a four digit number a b c d, counting from the left, positions 1 and 3 are odd, and positions 2 and 4 are even. The rule can be written as: (sum of digits in odd positions) minus (sum of digits in even positions) should equal 0 or ±11, ±22, and so on.

Step-by-Step Solution:
Step 1: Write the number in positional form. The digits are 5 (thousands place), 6 (hundreds place), M (tens place), and 4 (units place).Step 2: Identify odd and even positions from the left. Position 1: 5, position 2: 6, position 3: M, position 4: 4. Thus, odd positions are 1 and 3, and even positions are 2 and 4.Step 3: Compute the sum of digits at odd positions. That sum is 5 + M.Step 4: Compute the sum of digits at even positions. That sum is 6 + 4 = 10.Step 5: Apply the divisibility rule for 11. The difference must be 0, 11, or minus 11. So, (5 + M) - 10 must equal 0 or ±11.Step 6: Simplify the expression. (5 + M) - 10 = M - 5.Step 7: Set M - 5 equal to 0, 11, or -11 and check valid digit values. If M - 5 = 0, then M = 5. If M - 5 = 11, then M = 16, which is not a single digit. If M - 5 = -11, then M = -6, which is also invalid.Step 8: Conclude that the only possible valid digit is M = 5.

Verification / Alternative check:
Substitute M = 5 into the original number to obtain 5654. Now check divisibility by 11 either by performing short division or by re-applying the rule. Using the rule again: odd positions sum = 5 + 5 = 10, even positions sum = 6 + 4 = 10, and the difference is 10 - 10 = 0. Since 0 is a multiple of 11, 5654 is divisible by 11. This confirms that M = 5 is correct. A quick trial with any other option, such as M = 3, would give the difference as 3 - 5 = -2, which is not a multiple of 11, so those values are rejected.

Why Other Options Are Wrong:

• Option A (0): If M = 0, the number becomes 5604. Then odd positions sum = 5 + 0 = 5, and even positions sum = 6 + 4 = 10. The difference is 5 - 10 = -5, which is not a multiple of 11.
• Option B (1): If M = 1, the number becomes 5614. Odd sum = 5 + 1 = 6, even sum = 6 + 4 = 10, difference = 6 - 10 = -4, not a multiple of 11.
• Option C (3): If M = 3, the number becomes 5634. Odd sum = 5 + 3 = 8, even sum = 6 + 4 = 10, difference = 8 - 10 = -2, also not a multiple of 11.

Common Pitfalls:
A common mistake is to misunderstand the positions of digits, especially whether to count from the right or the left. The standard rule counts positions from left to right. Another mistake is to forget that the difference can be 0 or ±11, not just 0. Some candidates incorrectly attempt full long division, wasting precious exam time. Memorising and practising the divisibility rule for 11 will make such questions quick and efficient to solve.

Final Answer:
The digit M must be 5 for 56M4 to be completely divisible by 11.