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Math Help - 4x5 integer table

  1. #1
    Member u2_wa's Avatar
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    Tricky Integer Table

    Twenty different positive integers are written in a 4 X 5 table. Any two neighbours (numbers in cells with a common side) have a common divisor greater than 1. If n is the largest number in the table, find the smallest possible value of n.
    Last edited by u2_wa; August 26th 2012 at 10:55 AM.
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  2. #2
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    Re: 4x5 integer table

    I present you with the following solution.

    Spoiler:


    Let m the minimum such n. I came up with the following upper bound: m \leq 26 with the following solution to the puzzle

    \begin{bmatrix}3   & 6  & 15  & 25  & 5 \\24 & 9   & 18 & 20 & 10\\21 & 12 & 26 & 8 & 4\\7   & 14 & 22  & 16 & 2\end{bmatrix}

    If we inspect this table, we see that it satisfies the conditions for divisibility and the upper bound is correct. Now, if a prime p appears explicitly in the table, its neighbors must be divisible by that prime. If the prime appears in the interior, its neighbors must be at least the sizes 2p, 3p, 4p, 5p. If it appears on an edge, the neighbors are at least 2p, 3p, 4p. If it appears in a corner the neighbors are at least 2p, 3p. Using the lowest of these upper bounds 3p, in an optimal solution we must have that 3p \leq 25, or p \leq 26/3 \leq 9. That is, the only explicit primes in this table are at most 2,3,5 and 7. No other primes can appear explicitly in the table. As we can see, up to 26, the only numbers that do not appear in this table are 1, 11, 13, 17, 19 and 23. Since these numbers are not allowed in an optimal solution, the solution I provided must be optimal. Therefore m = 26.


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