# On a distant island, there is a class of 200 students; 33 are boys and 101 are girls. In what numerical system are they calculating? [closed]

On a distant island, there is a class of 200 students; 33 are boys and 101 are girls. In what numerical system are they calculating?

I know what numerical systems are pretty well, but this exercise semms weird to me. How can I even figure it out from the information given?

• Just try the various base.
– lulu
Commented Jun 18 at 16:03
• Well, in base $10$ we don't have $200$ equal to $33+101$ (assuming that everyone is either a boy or a girl, which I assume is the intent here). What about base $5$? or base $9$? Trial and error yields the solution rapidly.
– lulu
Commented Jun 18 at 16:06
• $3+1\equiv 0\pmod b$ Commented Jun 18 at 16:18
• No. I use base $10$. What is base $4$? Commented Jun 18 at 19:05
• Could be plain old base 10 if there are a whole lot of nonbinary people in the class. Commented Jun 19 at 1:30

The answer is base $$4$$, that is, assuming every student has to be a boy or a girl.

When we write something like $$123$$ in base $$b$$, what we really mean is $$1\cdot b^2+2\cdot b+3\cdot1$$.

So this question reduces to solving a quadratic equation, namely

$$2b^2=(3b+3)+(b^2+1)$$.

Rearranging, we find $$b^2-3b-4=0$$ and so we can find $$b=4$$ by factoring.

• or $b=-1$, which is an extraneous solution Commented Jun 18 at 16:24
• b=-1 is not necessarily extraneous. Perhaps the islanders use a weird version of base -1, where the rightmost digits are 1s, then -1, 1, -1, 1 etc. Thus "200" means there are 2 islanders, "33" means 0 boys, and "101" means 2 girls.
– Matt
Commented Jun 19 at 2:21
• Well, assuming boys or girls are the only options and the base is not negative, I did it like this: 200 > 134 so the base is < 10. Considering the last digits, 1+3 = 0 (i.e. 10) so the base is 4. Commented Jun 19 at 2:50

If you set it up as an addition problem $$\ 101\\ \underline{+ 33} \\ \ 200$$ You can see that $$1+3=10$$, so the base must be $$4$$

• Quick and neat! Commented Jun 19 at 3:08

$$200 - 101 = dd$$ in base $$d+1$$.

$$200 = 101 + 33$$, so $$200 - 101 = 33$$.

Thus, $$d = 3$$ and the base is $$4$$.