2019

The workings

Now that the year 2019 is halfway through, it is time to propose an amusement in which this number is the leading character. On this occasion, the idea is to separate the number into two sums, as if we were dividing the year into two parts, perhaps because of the end of the academic year.

The fun

Prove that if \(x\) and \(y\) are positive integers such that \(x + y = 2019\), then \(x \cdot y\) is not divisible by \(2019\). Prove that the same is true for any other square-free number other than \(2019\).

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Solution

We encourage the readers to try to solve the divertimento for themselves. Whether you succeed or not, you can always consult the solution in this link.