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 Geometric and Harmonic Harmony (Posted on 2014-12-15)
Find four positive integers A < B < C < D that satisfy both of these conditions:

(i) The geometric mean of A and D is 24, and:
(ii) The harmonic mean of A, B, C and D is 19.2

 No Solution Yet Submitted by K Sengupta Rating: 5.0000 (1 votes)

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 re(2): solution Comment 4 of 4 |
(In reply to re: solution by Steve Herman)

DM, Steve is correct! Here's why:

From (ii), 576 (1/B + 1/C) = 120 - 576 (1/A + 1/D)

i.e.,          576 (B + C)/BC = 120 - (A + D),    since AD = 576

The RHS of this equation is a positive integer, so BC must also be a factor of 576. From the list of possible factors, we conclude that only (A, B, C, D) = (8, 24, 36, 72) and (12, 16, 24, 28) are feasible solutions.

Great problem. KS!

 Posted by JayDeeKay on 2014-12-22 12:37:34

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