How to determine the base of a Number given a problem?

[dk702]

The problem is a follows

142 alpha 214 = 331
and
431 beta 123 = 303

where alpha and beta are unknow operators

I am pretty sure they are +,-,*,/

I know the radix (base) must be 5 or higher because 4 is present

By assuming the base to be 5 and converting to decimal, I discover beta = - and the base is indeed five. But if I convert the first part the first part to decimal useing base 5 the answer is wrong.

It is possible I copied the problem incorrectly

In general, my question is, how to determine the base to a number system given a problem similar to the one above.

 

[HallsofIvy]

The only way I could think of is to TRY various bases, various operations and SEE which one works. Somethime the best way to solve a problem is actually do all the "donkey work".

 

[Gokul43201]

If you are restricted to just the +/- operations, you can rewrite the problem in term of two unknowns (if alpha, beta are given to be distinct; else three unknowns). Let the base be 'b' :

$$(2+4b+b^2) + (-1)^n (4+b+2b^2) = 1+3b+3b^2$$

and $$(1+3b+4b^2) + (-1)^m (3+2b+b^2) = 3+3b^2~~~n,m ~\epsilon~ {0,1}$$

 

[Gokul43201]

Here's another thing to notice ...and this, in conjuction with the above type of method, gives you a solution.

<small number> alpha <large number> =<larger number>, all numbers positive

So alpha must be addition or multiplication. But the number of digits of LHS and RHS are the same (three), so it must be addition. By a similar reasoning, beta is subtraction.

Thus, you copied the problem down incorrectly.