Problem of the Week 1160

Many Pairs Dividing a Sum

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Solution

Problem 1160 was solved by Richard Boardman, Joseph DeVincentis, Mark Krusemeyer, and Ken Duisenberg.

Boardman's proof that 4 is optimal is nice and short:

For any 4 numbers, no more than 4 of the 6 possible pair sums can be factors. Let the 4 numbers be a, b, c, d in ascending order. Then c + d and b + d must exceed half the total and cannot be factors.

There are many sets of 4 numbers with 4 pairs as factors, e.g., 1, 5, 7, 11.

[Back to Problem 1160]

© Copyright 2012 Stan Wagon. Reproduced with permission.