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This is a 5x5 multiplication table.

    1   2   3   4   5
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1 | 1   2   3   4   5
2 | 2   4   6   8   10
3 | 3   6   9   12  15
4 | 4   8   12  16  20
5 | 5   10  15  20  25

Ordering all unique products by magnitude, you get:

5*5, 5*4, 4*4, 5*3, 4*3, 5*2, 3*3, 4*2, 3*2, 5*1, 4*1, 3*1, 2*1, 1*1
 25,  20,  16,  15,  12,  10,   9,   8,   6,   5,   4,   3,   2,   1

Given any two product pairs, could you predict which would have the greater product without performing the multiplication?

(3,2) or (5,1) ?

My first guess, whichever pair has a greater sum, is proven incorrect by the above example. Is there another test that can be done?

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You are asking to compare $ab$ with $cd$. You can ask whether $\frac ac$ is greater or less than $\frac db$, which lets you divide instead of multiply. You can also compare $\log a + \log b$ with $\log c + \log d$, which is easy if you have log tables around.

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  • $\begingroup$ Is there a way to do this without using division or log tables? $\endgroup$ – Adrian May 31 '16 at 4:28
  • $\begingroup$ @Adrian: I don't think so, particularly as the numbers get larger. You want the sum to be split evenly between the multiplicands, but how you trade off making one higher versus splitting more evenly is far from obvious. OP's example of $(1,5)$ vs $(2,3)$ shows the problem well, but we all know that $2 \times 3 \gt 1 \times 5$. How about $(12345, 6789)$ vs $(10101, 8297)$? $\endgroup$ – Ross Millikan May 31 '16 at 4:34

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