# Convert base-10 212 to base-9

1. Nov 19, 2009

### soopo

1. The problem statement, all variables and given/known data
How can you convert 212 in base-10 to base-9?

3. The attempt at a solution

There are holes in the sequence of figures when considering 212 in base-9.
There are 21 holes in the line. There also holes from 90 to 99 and from 190 to 199.
There are then 40 (= 23+10+10-1) holes in the line, since 198 belongs to the previous sets already.
This suggests me that the figure in base-9 is at least 242.
However, the correct answer is 248.

Last edited: Nov 19, 2009
2. Nov 19, 2009

### Staff: Mentor

21210 = 2*102 + 1*101 + 2*100

What you want to do is write this number as the sum of multiples of powers of 9, which are 1, 9, 81, 243, and so on.

How many 81's (= 92) are there? That will be you digit in the 92 place.
If there's a remainder after subtracting of the 81's, how many 9's are there in that remainder? That will be your digit in the 9's place.
If there's a remainder (and it better be 5) that will be your 1's place.

If you're converting 21210 to its base-9 equivalent, 248 IS NOT the correct answer. I don't understand what you're talking about with holes in the line.

3. Nov 20, 2009

### icystrike

The answer should be $$(255)_{9}$$ . Correct me if i'm wrong.
212=207+5=23*9+5
23=2*9+5
2=0*9+2

Combining all together,

212=9(2*9+5)+5=2*9²+5*9+5
Suggesting 212=$$(255)_{9}$$

Last edited: Nov 20, 2009
4. Nov 20, 2009

### ideasrule

Yes, 255 is right. That's easy enough to check: 2*9^2+5*9+5=212.

5. Nov 20, 2009

### HallsofIvy

Staff Emeritus
The quick way to do this is: divide 212 by 9. 9 divides into 212 23 times with remainder 5. 9 divides into 23 twice with remainder 5: 21210= 2559.

That is because saying that 9 divides into 212 23 times with remainder 5 means that 212= 9(23)+ 5. Since 9 divides into 23 twice with remainder 5, 212= 9(9(2)+ 5)+ 5= 2(92)+ 5(9)+ 5.