13 thoughts on “Base Systems – Examples

  1. Sir i am unable to understand logic behind solving question 3.
    see if i take an example as to write digits only with 1,2,3 then 10th digit will be 31 (1, 2,3,11,12,13,21,22,23,31)
    but in similar way if i say number 10 in base 3 than it will be 101 though it is obvious this can’t be the answer but this arises ambiguity in answer.

  2. That is because the answer contains a 0 (and you had only allowed digits 1, 2 and 3 in the original!) So (10)base3 is the same as 3. (Yes it gets a bit complicated if the digit 0 does not exist!).

    Let me take another example. Suppose I have 5 digits available, 0, 1, 3, 6, 8 and I want to know the 108th number possible. Here we can write it in base 5 (but notice that 0, 1, 3, 6 and 8 respectively correspond to the digits 0, 1, 2, 3 and 4 in base 5!). (108)5 = 413, so 4, 1 and 3 correspond to 8, 1 and 6 and hence the required number will be 816.

    Now when the digit 0 itself is missing, the confusion will arise if and only if 0 is present in the converted value…(I could demonstrate it to you on an excel sheet 🙂 But not sure how to put it in words…as in, a rigorous proof!) For all other numbers it will give the expected result.


    • Sir, why did you replace 4,1, and 3 here with 8,1, and 6 respectively
      but not 2,4, and 2 with 3,5, and 3 in question 3.

    • Hello Sir, Please check if my understanding is correct.
      If we get a final result(after converting to the required base) such that the digits of the number obtained (242 in the question 3) are present in the number system given in the question, then that number is the answer. Otherwise, we will have to find the corresponding digits if the result obtained is not using the digits of the base(as you explained above (108)5 = 413 ~ 816).

      For eg, if our number system uses 2,3,5,9 and we need to find the 50th number in this base. Then (50)5 = 200 and since 0 is not present in the current base. we see that 0->2 and 2->3.. hence the number will be 322. Is this correct?

      Scenario 2: Same Number system (2,3,5,9).. and we need to find 67th number in this base. Then (67)5 = 232. Since all the numbers are present, 232 will be the answer(what i understood according to solution of question 3)? if not we find that 2 corresponds to 5 here and 3 corresponds to 9 here so the answer will be 595.
      However, you have simply taken the number 242 as the answer and not converted it as per the corresponding base system.. 242 -> 353 ?

      Please elucidate utterly confused.

      • So either Scenario 1 or Scenario 2 is incorrect. Since if 50th number is 322, 67th number cannot be 242 or is it 353 ?…

        In total I have 3 answers for 2 questions 🙂 and unsure which one is right one. 😦

      • I am afraid you are wrong. You need to maintain an equivalence between the digits allowed and the digits in the base system, and it becomes more complicated if there is no 0 in the allowed digits. In the case you took where the digits allowed are 2, 3, 5, 9, there is no zero so straightaway it becomes a little bit more messy (who am I kidding…it becomes a lot more messy actually).

        The case I took had 0 as well. The digit 9 was the only one missing so it would directly give me the correct digits. Had the allowable digits been 0, 2, 3, 5 and 9 then I would use base 5 and say that (0, 1, 2, 3, 4) in base 5 corresponds to (0, 2, 3, 5, 9) in the final answer. So supposing I were to get (201234) as the base 5 answer I would say the final answer would be (302359).


      • Sir, but in Question 3, the number 0 is not present in the base system, still you arrived at the correct answer. So if the number 0 is not present in the base and it does not come up in the conversion, the converted number can be assumed to be final result?

        And if 0 is present, then we will find the corresponding numbers? is this correct? Please clarify

    • From what I understood, by experimenting in excel, even if 0 is present we start counting from the next greater digit…but in case 0 is not present then we start counting from whatever smallest digit is present. That’s why the corresponding method doesn’t work when 0 is skipped, like in the question you have explained.
      Hope my understanding is correct.

    • Nishant, try it for yourself. write a few 3-digit numbers in base 5, say 100, 111, 234, 444 and convert them into base 10. You will find that the smallest number is (100)5 = 25 in base 10 while the largest is (444)5 = 124 in base 10. These numbers are 5^2 and 5^3-1…


  3. Hi,
    In the example(2324×1013) base 7, completely divisible with 6, why is the addition of numbers taken and solved with base 10? 16+x=18 ?
    Also tell me the answer to: (32323232….150 digits) base 9. Remainder when divided by 8. Kindly share the approach to solve through sum of the digits.

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