A Problem of Divisibility

Arrange n×m digits aij (i = 1, ..., n; j = 1, ..., m) in a rectangular array. If read left to right, every row then represents a decimal integer, and so does every column if read from top downwards. In all, there are n + m integers. Let p be relatively prime to 10. (Two integers are relatively or mutually prime if they have no common divisors except 1. They are also called coprime.) Being prime to 10 means that p is odd but is not divisible by 5. 3, 7, 9, 11, 21 are examples of such integers.

Here's a problem. Assume it's known that all n + m numbers in the matrix but one are divisible by p. Prove that the remaining number is bound to be divisible by p as well.

The applet below helps you verify that this is so. All digits are clickable so that you can modify the matrix to your liking.


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Problem of Divisibility


What if applet does not run?

Proof

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