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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.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (1 votes)

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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

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