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

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In summary, the problem involves two equations with unknown operators, alpha and beta, and a base of 5 or higher. By assuming the base to be 5 and converting to decimal, it is discovered that beta is subtraction and the base is indeed 5. However, when converting the first equation to decimal using base 5, the answer is wrong, indicating that the problem was copied incorrectly. The best way to solve the problem is to try various bases and operations, and see which one works. Another observation is that alpha must be addition and beta must be subtraction, indicating that the problem was copied incorrectly.
  • #1
dk702
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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.
 
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  • #2
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".
 
  • #3
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' :

[tex](2+4b+b^2) + (-1)^n (4+b+2b^2) = 1+3b+3b^2[/tex]

and [tex] (1+3b+4b^2) + (-1)^m (3+2b+b^2) = 3+3b^2~~~n,m ~\epsilon~ {0,1}[/tex]
 
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  • #4
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.
 
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1. How do you identify the base of a number in a given problem?

To determine the base of a number in a problem, you need to look for clues such as the presence of a subscript or a decimal point. The subscript indicates the base of the number, while the decimal point signifies the use of base 10.

2. What are some common bases used in mathematical problems?

The most frequently used bases in mathematical problems are base 2 (binary), base 8 (octal), base 10 (decimal), and base 16 (hexadecimal). These bases are used in different contexts, such as computing, computer programming, and digital electronics.

3. How can I convert a number from one base to another?

To convert a number from one base to another, you can use the standard conversion formula:
New Number = (Old Number) x (Old Base / New Base).
For example, to convert a number from base 10 to base 2, you would multiply the number by 2 and then divide by 10.

4. Can you solve a problem without knowing the base of the numbers?

Yes, you can solve a problem without knowing the base of the numbers by treating them as variables. You can then use algebraic techniques to simplify the equation and find the solution. However, knowing the base can help you understand the problem better and may lead to a more efficient solution.

5. How does understanding the base of a number help in problem-solving?

Understanding the base of a number can help in problem-solving by providing a framework for solving the problem. It can also help in identifying patterns and relationships between numbers and can guide you towards the correct solution method. Additionally, knowing the base can help you avoid mistakes and errors in calculations.

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