Number Base System: Solving the Radix/Base

[tnutty]

Homework Statement

This really isn't a calc question so forgive me.

Determine which base(radix) is used in the following operation :

a ) 1234 + 1234 + 1234 + 1234 = 11101
b ) 19 * 18 = 297

I know the answer to a but am not sure how to get it.
What I see is the for a) the answer has to be greater than base 4, and for b) it has
to be radix >=10.

Is there some tricks that could help me figure out what base is in general a expression is on?

 

[Mark44]

For a, yes, the base has to be larger than 4, so use the fact that 4 + 4 + 4 + 4 = base + 1 or maybe 2* base + 1. Check the addition in a few bases (probably not an even base), starting with 5.

For b, I agree that the base is probably larger than 10. Again try a few bases and see if you can find one for which 9 * 8 = base + 7. Since you're getting a unit's place of 7, it's probably not an even base.

 

[tnutty]

For A its 5, but I can't seem to figure out part B.

This is what I got :

let "r" be the base.

(9*8)r = ?
-----------

r = x | result = y | (1|0)
-----------------
11 | 68 | 11 * 6 = 66, left over 2. False
13 | 57 | 13 * 4 = 52, left over 5, False

And it goes on.

What I did was 9*8 = 72. In base 10. Then divide 72 by different bases to see if it matched up.

wait is r = 13 correct?

 

[Elucidus]

For A its 5, but I can't seem to figure out part B.

This is what I got :

let "r" be the base.

(9*8)r = ?
-----------

r = x | result = y | (1|0)
-----------------
11 | 68 | 11 * 6 = 66, left over 2. False
13 | 57 | 13 * 4 = 52, left over 5, False

And it goes on.

What I did was 9*8 = 72. In base 10. Then divide 72 by different bases to see if it matched up.

wait is r = 13 correct?

Yes, the radix is 13 in part (b). Can you show why?

--Elucidus

 

[Mark44]

Keep in mind that the numbers in the product are in the same base as the answer, so what you have for b is 19_r * 18_r = 297_r.

What I did was to make an educated guess as to the base, and then convert all three numbers into their base-10 equivalents and check the multiplication in that more familiar base.

 

[tnutty]

Keep in mind that the numbers in the product are in the same base as the answer, so what you have for b is 19_r * 18_r = 297_r.

What I did was to make an educated guess as to the base, and then convert all three numbers into their base-10 equivalents and check the multiplication in that more familiar base.

I was thinking of doing that as well.

Lets see :

19r * 18r = 297r

I know its not base 10, but from (base 10) 9*8 = 72
Now I can use this to convert into other base and check accordingly.

13*5 = 65;
72 - 65 = 7

So 7 is left over and even number of 13 is carried onto the next step. Ah, its just
like you said, look for "base + 7".

It would be nice if someone could offer alternative solution.