Divisibility Rule for 11 Explained

To determine if a number is divisible by 11, we use a specific rule. This rule involves the alternating sum of the digits. We sum the digits in the odd positions (starting from the rightmost digit as position 1) and subtract the sum of the digits in the even positions. If the result is 0 or any multiple of 11 (like 11, 22, -11, -22, etc.), the original number is divisible by 11.

Applying Divisibility Rule to N01M22

The given 6-digit number is N01M22.

Let's identify the digits and their positions from right to left:

• Position 1 (Odd): 2
• Position 2 (Even): 2
• Position 3 (Odd): M
• Position 4 (Even): 1
• Position 5 (Odd): 0
• Position 6 (Even): N

Now, let's calculate the sum of digits at odd positions and even positions:

• Sum of digits at odd positions = (Digit at pos 1) + (Digit at pos 3) + (Digit at pos 5)
  = $2 + M + 0 = M + 2$
• Sum of digits at even positions = (Digit at pos 2) + (Digit at pos 4) + (Digit at pos 6)
  = $2 + 1 + N = N + 3$

According to the divisibility rule for 11, the difference between these sums must be a multiple of 11.

Difference = (Sum of odd positioned digits) - (Sum of even positioned digits)

Difference = $(M + 2) - (N + 3)$

Difference = $M + 2 - N - 3$

Difference = $M - N - 1$

So, for N01M22 to be divisible by 11, the expression $M - N - 1$ must be a multiple of 11.

We can write this as:

Finding the Relation between M and N

Rearranging the equation, we get:

We know that M and N are digits. Since N is the first digit of a 6-digit number, N cannot be 0. Therefore:

• $1 \le N \le 9$
• $0 \le M \le 9$

Let's find the possible range for the difference $M - N$:

• Maximum value of $M - N$ is when M is maximum (9) and N is minimum (1): $9 - 1 = 8$.
• Minimum value of $M - N$ is when M is minimum (0) and N is maximum (9): $0 - 9 = -9$.

So, the difference $M - N$ must be within the range $[-9, 8]$.

Now let's test values of $k$ in the equation $M - N = 11k + 1$:

• If $k = 0$: $M - N = 11(0) + 1 = 1$. This value (1) is within the possible range $[-9, 8]$. This gives a possible relation $M - N = 1$.
• If $k = 1$: $M - N = 11(1) + 1 = 12$. This value (12) is outside the range $[-9, 8]$.
• If $k = -1$: $M - N = 11(-1) + 1 = -10$. This value (-10) is outside the range $[-9, 8]$.
• For any other integer value of $k$, the result for $M - N$ will be even further outside the range $[-9, 8]$.

Therefore, the only possible relation between M and N that satisfies the divisibility rule for 11 is $M - N = 1$.

Analyzing the Options

Let's check which option matches our derived relation $M - N = 1$.

Conclusion

Based on the divisibility rule for 11 and the constraints on the digits M and N, the only possible relation is $M - N = 1$. This matches Option 1.