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Four Digit Trial (Posted on 2014-06-05) Difficulty: 3 of 5
N is a 4-digit positive integer such that the sum of the four digits of N equals the product of the first two digits of N and also equals the product of the last two digits of N.

Find all N's and prove there are no others.

No Solution Yet Submitted by K Sengupta    
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Solution Finding all N's Comment 3 of 3 |

Solution:

1. It is easily found  that there cannot exist an abcd solution with three or four digits distinct.

2. A generic abab solution will generate the

abba,  baba   ,  baab    solutions. <br>

3. Write   2*(a+b)-a*b  as  a =2*b/(b-2),  which produces integer values for b=3  or 4 or 6 only.

4. We get generic answers:  (a,b ) = (3,6);  (4,4 );  (6,3 ).

5.The set of all valid answers is: (4,4,4,4); (3,6,6,3);  (3,6,3,6);  (6,3,6,3); );(6,3,3,6);


 General remark:   Finding all Ns  constitutes

a full proof  there   are no  others.                           

 


  Posted by Ady TZIDON on 2014-06-06 04:33:13
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