 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  My favorite numbers II (Posted on 2009-12-21) Determine all possible sextuplets (A, B, C, D, E, F) of positive integers, with A ≤ B ≤ C, and, D ≤ E ≤ F and, A ≤ D, that satisfy both the equations: A+B+C = D*E*F and, A*B*C = D+E+F.

Prove that these are the only sextuplets that exist.

 No Solution Yet Submitted by K Sengupta Rating: 4.0000 (1 votes) Comments: ( Back to comment list | You must be logged in to post comments.) The start of a proof... Comment 4 of 4 | Lemma 1)
if x, y, z are all >= 2,
then xyz > x + y+ z

This follows because
xyz >= 4x
xyz >= 4y
xyz >= 4z
Average the 3 together, and we get
xyz >= (4/3)(x + y + z)
Therefore xyz > x + y+ z

Step 1)
If A >= 2,
then D >= 2
DEF > D+E+F, (from lemma 1)
D+E+F = ABC
ABC > A+B+C  (from lemma 1)
A+B+C = DEF

implies DEF > DEF,

which is a contradiction,
so A = 1

 Posted by Steve Herman on 2009-12-23 10:20:36 Please log in:
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