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):
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.
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| Looking at the last digits of these numbers, |
| you notice that neither 2 nor 3 nor 4 is the base. |
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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.
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It is nice entertainment problem with non-standard setup.
And making right setup is the most engaging/enlightening part of its solution.
