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Find the area (Posted on 2011-04-08) Difficulty: 3 of 5
If RE*ER= AREA evaluate the AREA .

Please evaluate using pencil and paper only.

See The Solution Submitted by Ady TZIDON    
Rating: 4.0000 (3 votes)

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Solution solution Comment 5 of 5 |
Let x = RE
Let y = ER

AREA = 1001*A + 10*x
RE*ER = x*y
x*y = 1001*A + 10*x
x*(y - 10) = 1001*A

The prime factorization of 1001 is 7*11*13, therefore we have the following:
  1. x=                7 & y= 11*13*A+10   = 143A+10
  2. x=               11 & y= 7*13*A+10    =  91A+10
  3. x=               13 & y= 7*11*A+10    =  77A+10
  4. x= (7*11)    =   77 & y=                 13A+10
  5. x= (7*13)    =   91 & y=                 11A+10
  6. x= (11*13)   =  143 & y=                  7A+10
  7. x= (7*11*13) = 1001 & y=                   A+10
  8. x=               7A & y= (11*13)+10   =     153
  9. x=              11A & y= (7*13)+10    =     101
  10. x=              13A & y= (7*11)+10    =      87
  11. x= (7*11*A)  =  77A & y= 13+10        =      23
  12. x= (7*13*A)  =  91A & y= 11+10        =      21
  13. x= (11*13*A) = 143A & y= 7+10         =      17
  14. x=                A & y=(7*11*13)+10  =    1021

As x represents the two digit cryptarithmetic number RE, and y represents the two digit cryptarithmetic number ER, such that R and E are different digits and neither can be 0, we eliminate most of the above possibilities. If we let A=1, the minimum possible value for the cryptarithmetic digit, we can eliminate the remaining possibilities but for [10.]. Thus y represents the two digit number 87.
The two digit cryptarithmetic number RE must then be 78.

Dividing by 13 for each side of the equation 13*A = 87, we find A = 6.

We can validate by substitution:
x*(87 - 10) = 1001*(6) => x = 6006/77 = 78
and the equation 78*87 = 6786.


  Posted by Dej Mar on 2011-04-09 13:47:06
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