$$23 + 25 = 51 $$ What base is used in the above addition operation ?
I have 2 methods to do this
Method 1 : Through equations
assume base be a
$$23_a + 25_a = 51_a $$
$$2a + 3 + 2a + 5 = 5a +1 $$
solving this we get a as 7
Method 2 : Just logical reasoning
as the digit in $ a^1 $'s place is $5$ instead of $4$ (because $2 + 2 =4$). This means that a carry came from addition of digits in $ a^0 $'s place . This means $ 5 + 3 = 11 $ here . so it means $ 5 + 1 = 6$ (Just finding addition which does not result in carry) . so base is $7$ because after $6$ adding $1$ leads to $10$ , then another $1$ added leads to $11$ .
No problem with the above methods . But these methods are usable and easy to visualise when numbers are small and when simple operations like addition are involved . The methods become tedious to follow when number's are large and when product or division operations are given .
So is there a better way to approach these kind of problems where an operation is given and asked to find the base in which the operation was performed .
To be more specific and precise,Is there way to tell the base just by comparing the operation's result (in the unknown base) and value it will get in decimal system(base 10) and tell the base in which the operation was performed ?
That is as $$ 23_a + 25_a = 51_a $$
$$ 23_{10} + 25_{10} = 48_{10} $$
Comparing $48_{10}$ (result in decimal system) and $51_a$ (result in base a system) will I be able to conclude that base a is nothing but $7$ .
I am quiet jealous in wanting to find it by seeing its result in decimal system because in general we are more used to decimal system than any other base .
Thanks
Best Answer
In general, you will find that you get expressions which convert to polynomial equations in the base. If you know the base is an integer, you will then be able to use the rational root theorem to test the limited number of possible bases. Some of these possibilities are likely to be eliminated by other considerations.