# Number base system

1. Sep 15, 2009

### tnutty

1. The problem statement, all variables and given/known data

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

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

2. Sep 15, 2009

### Staff: Mentor

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.

3. Sep 15, 2009

### 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?

Last edited: Sep 15, 2009
4. Sep 15, 2009

### Elucidus

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

--Elucidus

5. Sep 16, 2009

### Staff: Mentor

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

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.

6. Sep 16, 2009

### tnutty

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.