# Thread: Finding bases from arithmetic ops

1. ## Finding bases from arithmetic ops

Determine the base of the following operation:

1) 302 / 20 = 12.1

and

2) 1234 + 5432 = 6666

Assume that the base of the system is x.

On the 1st problem, I would think the answer would be base 13, since the division in base 10 is 15.1. Is this correct?

On the 2nd problem, I would think base 10, but I don't think there is enough information given, since there are no carries.

2. Originally Posted by aaronrj

On the 1st problem, I would think the answer would be base 13, since the division in base 10 is 15.1. Is this correct?
You are correct to say it is base 10 but you cannot imply base 13 given simply this. The need to show it is base 13

Originally Posted by aaronrj

On the 2nd problem, I would think base 10, but I don't think there is enough information given, since there are no carries.

By inspection you can see it is base 10.

3. ## has no solution?

Originally Posted by pickslides
You are correct to say it is base 10 but you cannot imply base 13 given simply this. The need to show it is base 13
By inspection you can see it is base 10.
is the question
302/20=12.1 proper?

i tried to take x as the base and trace the division algebraically:
( number 302 implies x>3 for the digit 3 appears in some number-----inference 1)

x + 2 + x^-1
----------------------------------
2x + 0 ) 3x^2 + 2
, :
, :

is what i expected to trace.
Following,
x
----------------------------------
2x + 0 ) 3x^2 + 2
, 2x^2
, ---------------
, x^2 comparing with ordinary arithmetic way,
x^2 must be lesser than 2x ( otherwise the quotient would be 2x or more ).

However, x^2 < 2x is true only when x=1 which is a contradiction with inference 1

what is going wrong here?
:

4. ## ok

To find the base of 302 / 20 = 12.1, could I write

3 * x^2 + 0 * x^1 + 2 * x^0

Then write 12.1 as

1 * x^1 + 2 * x^0 + 1 * x^-1

and then solve for x?

Also, how would I show that 1234 + 5432 = 6666 is base 10?

5. Originally Posted by aaronrj
Determine the base of the following operation:
1) 302 / 20 = 12.1
...

Assume that the base of the system is x.

On the 1st problem, I would think the answer would be base 13,
Is this correct?
...
in base 4 this division is correct:
302 / 20 = 12.1

6. Originally Posted by integerfan
is the question
302/20=12.1 proper?

i tried to take x as the base and trace the division algebraically:
( number 302 implies x>3 for the digit 3 appears in some number-----inference 1)

x + 2 + x^-1
----------------------------------
2x + 0 ) 3x^2 + 2
, :
, :

is what i expected to trace.
Following,
x
----------------------------------
2x + 0 ) 3x^2 + 2
, 2x^2
, ---------------
, x^2 comparing with ordinary arithmetic way,
x^2 must be lesser than 2x ( otherwise the quotient would be 2x or more ).

However, x^2 < 2x is true only when x=1 which is a contradiction with inference 1

what is going wrong here?
:
The idea of a base is positional notation.
302
in decimal it is

$3 \cdot 10^2 + 0 \cdot 10^1 + 2 \cdot 10^0
$

in base x it is

$3 \cdot x^2 + 0 \cdot x^1 + 2 \cdot x^0
$

Try it that way.

$( 1x^3 + 2x^2 +3x^1 + 4x^0 ) + ( 5x^3 + 4x^2 + 3x^1 + 2x^0 ) = 6x^3 + 6x^2 + 6x^1 + 6x^0$

Solve for x.

.

7. Originally Posted by aaronrj
Determine the base of the following operation:

2) 1234 + 5432 = 6666

On the 2nd problem, I would think base 10, but I don't think there is enough information given, since there are no carries.
All you can say is that the base is greater than 6. I've thought about this and I can't see how you can get any more precise than this. Anyone care to show me wrong? It's bewildering.

,

,

### 1234 5432=6666 bases

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