Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.

Applying the Rule

Consider the number $N84M69$. Let's find the sums of digits at odd and even places, counting from the right:

• Sum of digits at odd places (1st, 3rd, 5th): $9 + M + 8 = M + 17$.
• Sum of digits at even places (2nd, 4th, 6th): $6 + 4 + N = N + 10$.

Calculating the Difference

The difference is $(M + 17) - (N + 10) = M - N + 7$.

This difference must be a multiple of 11.

Finding Possible Relations

Since $N$ and $M$ are digits, $0 \le N \le 9$ and $0 \le M \le 9$.

• The possible range for $M - N$ is $[-9, 9]$.
• The possible range for the difference $M - N + 7$ is $[-9 + 7, 9 + 7]$, which is $[-2, 16]$.

The multiples of 11 within $[-2, 16]$ are $0$ and $11$. This leads to two possible cases:

1. Case 1: $M - N + 7 = 0$

   Solving for $M - N$, we get $M - N = -7$. This relation is possible (e.g., if $N=7, M=0$).

2. Case 2: $M - N + 7 = 11$

   Solving for $M - N$, we get $M - N = 4$. This relation is also possible (e.g., if $N=1, M=5$).

Conclusion

The question asks for a possible relation. The relation $M - N = -7$ derived from Case 1 is one such possibility.