Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $15$, it has the same three digits, in reverse order. What is $N$? (Express your answer in decimal.)

Answer 1

Let $N = abc$, where $a$, $b$, and $c$ are digits when expressed in base $7$. Then $N = 49a + 7b + c$.

Switching the digits and expressing in base $15$, we have $N = 15^2 c + 15b + a$. Setting these two expressions equal, we have $49a + 7b + c = 225c + 15b + a$, or $48a + 8b = 224c$.

Since $a$, $b$, and $c$ are digits (less than 7), we see that $a = 4$ and $b = 3$. Substituting these in, we have $192 + 24 = 224c$, so $c = \frac{216}{224} = \frac{27}{28}$, which means $c = 1$.

Thus, $N = 49(4) + 7(3) + 1 = \boxed{198}$.