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My favorite numbers (Posted on 2005-07-22) Difficulty: 3 of 5
Find 4 different positive integers A,B,C,D for which: A+B = C*D and A*B = C+D How many sets of 4 numbers can you find? Prove that only those sets exist.

No Solution Yet Submitted by Jurgen    
Rating: 3.2000 (5 votes)

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Solution Solution | Comment 5 of 13 |

We know that all the numbers different positive integers, so there is either one 1 or none.

If there were none, then A+B<A*B, C+D<C*D so A+B<C+D<A+B which is a contradiction, so one of the numbers must be a 1. Since all the numbers are communitive (they can be switched around), I will just say A is 1.

This means B+1=C*D and B=C+D. Substituting in for B gives C+D+1=C*D, and if C>2, D>2 then C+D+1<C*D, so C (or D) must be 2. This means 3+D=2D and D=3, so B=5.   This means Nosher's set is the only set that meets the conditions in the problem.

  Posted by Gamer on 2005-07-22 14:23:58
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