SOLUTION: The correct addition 13 + 24 + 43 + 44 = 102 looks strange because it is in another base. The base is

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Question 1199769: The correct addition 13 + 24 + 43 + 44 = 102 looks strange because it is in another base. The base is
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Possibly the fastest way to solve the problem is by trying different bases and finding the one that works. But you can get better practice in problem-solving skills by solving the problem using logical reasoning.

(1) The sum of the units digits in base 10 is 14; since the last digit in the sum in the unknown base is 2, the unknown base must be an integer that is a divisor of 14-2 = 12.
(2) The addition in base 10 gives the result 124; since the addition in the unknown base gives a sum 102 using smaller digits, the base is greater than 10.

The only integer that satisfies both conditions is 12.

ANSWER: 12
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.

    +--------------------------------------------------------+
    |     Looking at the last digits of these numbers,       |
    |  you notice that neither 2 nor 3 nor 4 is the base.    |
    +--------------------------------------------------------+


Let the base be "b".  Then we can write this equation

    (b+3) + (2b+4) + (4b+3) + (4b+4) = .


Simplify and write in standard form of quadratic equation

    11b + 14 =  + 2

     - 11b - 12 = 0


Solve by factoring

    (b-12)*(b+1) = 0


The appropriate solution/answer is  b= 12.

Solved.

------------------

It is nice entertainment problem with non-standard setup.

And making right setup is the most engaging/enlightening part of its solution.



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