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Divide the expression, get triplets (Posted on 2007-05-19) Difficulty: 3 of 5
Determine all triplets of integers (a, b, c) satisfying 1 < a < b < c such that abc - 1 is a multiple of (a - 1)(b -1)(c - 1)

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Some initial analytics (spoiler) | Comment 2 of 4 |
a) By inspection, abc - 1 > (a-1)*(b-1)*(c-1)

b) But abc - 1 is a multiple of (a-1)*(b-1)*(c-1).
    Therefore, abc - 1 >= 2*(a-1)(b-1)(c-1),
        since abc - 1 <> (a-1)*(b-1)*(c-1)

    Adding 1 to each side and dividing by abc gives
    1 >= 2*((a-1)/a)*((b-1)/b)*((c-1)/c) + 1/abc

    But if a >= 4, then b >= 5 and c>= 6
    And (a-1)/a increases as a increases.
   
    At a minimum, the expression on the right =
    2*(3/4)*(4/5)*(5/6) + 1/abc = 1 + 1/abc, which is > 1

    So, a cannot be >=  4.

c) Therefore, a can only be 2 or 3

Edited on May 19, 2007, 8:31 pm
  Posted by Steve Herman on 2007-05-19 17:08:03

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