Problem 3 [10 pts]: Ms. Decibin the detective observes the following addition eq

Question

Problem 3 [10 pts]: Ms. Decibin the detective observes the following addition equation written in a top secret document At first she is very confused because neither of the "obvious" choices, 4, or 5, seem to satisfy the equation. After some thought, she realizes that a different base has been used to express the integers. Since 7 seems to be a digit she infers that the base is 8 or larger. Can you help Decibin to solve for the secret digit z as well as the unknown base b. The same base b was also probably used in other such equations, so this will make her task very easy! To be more specific find the value of the base b and the digit z. (It may not be one of 0-9 if b > 10.) Show all steps of your reasoning. Assume, as usual, that for bases larger than 10 one uses digits 0-9 as usual and the later digits are A,B, C,....

Explanation / Answer

Hello ,

First of all we need to understand , how addition of numbers works in different bases (there is a common algo );-

1.Start with writing both the numbers below each other , digit wise.

2.set initial carry=0 and now begin from right most digits and add them (in this case x+7) and plus carry

3.If the sum is less than the base , move on to next digits in the left.

4. It it is not , then subtract the base from sum, put down the result and set carry = 1.

5.If you are not out of digits, move to the next one to the left and go back to the step 2.

6. If the carry is not zero , put it down as the left most digit of the sum.

7. Stop.

So, if we now see the question , it is already inferred that base is greater than 8.

In the same way we can assume that x < base (because if x > base , then digit by digit addition will not be possible.)

So, making assumptions as follows-

Applying the algo here - assume x =5 and base =8

x+7 =12 (carry=1 and result=4) -- which is not correct according to our equation.

Moving on , lets assume x=4 and base=8

4+7=11 (carry=1 and result=3) -- again not correct.

Similarly , if we try , x=3 and base =9

3+7= 10 (10>9 , so carry =1 and result =1)-- correct

moving on to left digits carry + 2 +3=6 and carry=0 - correct again!!

Now to left most digits 3+3=6 -- not correct :(

so, we keep making assumptions like this--

Lets try base=12 and x=6

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