# Advancing a Millennium Problem

## Outline Mathematics

Algebra, Word Problems

Find all solutions in positive integers to

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Copyright © 1996-2017 Alexander Bogomolny

Find all solutions in positive integers to 2015=20·x+15·y.

### Solution

The problem is a modification of Problem #78 from David Singmaster's book Problems for Metagrobologists:

When the Millennium was approaching and the Dome and the Jubilee Line were still in the state of chaos, it was clear that we needed the extra year available as the next Millennium really didn't start until the beginning of 2001. This has very little to do with our problem except that it involves dates! Reading an 18C book, I was inspired to ask how many solutions are there to

I modified the problem and adapted David's method for solving his problem.

If we relaxed the requirements of the problem for a moment, it should make it easier to solve it, what do you think? Right,Right,Wrong,Don't know. Let's see. Is it easier to find a solution to the equation

What we are looking for then are numbers N such that

20·(403-3N) + 15·(-403+4N) = 20·403 + 15·(-403).

We thus require 403-3N ≥ 0 and 403-4N ≤ 0. The two inequalities are satisfied by

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