AIME 2018 II · 第 3 题

Solution 1

The first step is to convert $36_{b}$ and $27_{b}$ into base-10 numbers. Then, we can write

$$36_{b} = 3b + 6$$ and

$$27_{b} = 2b + 7$$ . It should also be noted that $8 \leq b < 1000$.

Because there are less perfect cubes than perfect squares for the restriction we are given on $b$, it is best to list out all the perfect cubes. Since the maximum $b$ can be is 1000 and $2$ • $1000 + 7 = 2007$, we can list all the perfect cubes less than 2007.

Now, $2b + 7$ must be one of

$$3^3, 4^3, ... , 12^3$$ . However, $2b + 7$ will always be odd, so we can eliminate the cubes of the even numbers and change our list of potential cubes to

$$3^3, 5^3, 7^3, 9^3\text{, and }11^3$$ .

Because $3b + 6$ is a perfect square and is clearly divisible by 3, it must be divisible by 9, so $b + 2$ is divisible by 3. Thus the cube, which is

$$2b + 7 = 2(b + 2) + 3$$ , must also be divisible by 3. Therefore, the only cubes that $2b + 7$ could potentially be now are $3^3$ and $9^3$.

We need to test both of these cubes to make sure $3b + 6$ is a perfect square.

$\textbf{Case 1:}$ If we set

$$3^3 = 2b + 7$$ so

$$b = 10$$ . If we plug this value of b into $3b + 6$, the expression equals $36$, which is indeed a perfect square.

$\textbf{Case 2:}$ If we set

$$9^3 = 2b + 7$$ so

$$b = 361$$ . If we plug this value of b into $3b + 6$, the expression equals $1089$, which is $33^2$.

We have proven that both $b = 10$ and $b = 361$ are the only solutions, so

$$10 + 361 = \boxed{371}$$

Solution 2

The conditions are:

$$3b+6 = n^2$$ $$2b+7 = m^3$$ We can see $n$ is multiple is 3, so let $n=3k$, then $b= 3k^2-2$. Substitute $b$ into second condition and we get $m^3=3(2k^2+1)$. Now we know $m$ is both a multiple of 3 and odd. Also, $m$ must be smaller than 13 for $b$ to be smaller than 1000. So the only two possible values for $m$ are 3 and 9. Test and they both work. The final answer is $10 + 361 =$ $\boxed{371}$. -Mathdummy

Solution 3

As shown above, let

$$3b+6 = n^2$$ $$2b+7 = m^3$$ such that

$$6b+12=2n^2$$ $$6b+21=3m^3$$ Subtracting the equations we have

$$3m^3-2n^2=9 \implies 3m^3-2n^2-9=0.$$ We know that $m$ and $n$ both have to be integers, because then the base wouldn't be an integer. Furthermore, any integer solution $m$ must divide $9$ by the Rational Root Theorem.

We can instantly know $m \neq -9,-3,-1,1$ since those will have negative solutions.

When $m=3$ we have $n=6$, so then $b=10$

When $m=9$ we have $n=33$, so then $b=361$

Therefore, the sum of all possible values of $b$ is

$$10+361=\boxed{371}.$$