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 $12$, it has the same three digits, in reverse order. What is $N$? (Express your answer in decimal.)

Question

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 $12$, it has the same three digits, in reverse order.  What is $N$?  (Express your answer in decimal.)

GPT 3.5 · Answer

Let $N = 7^2a + 7b + c = 12^2c + 12b + a$, where $a,b,c$ are the digits of $N$ in base $7$. Because $N$ has three digits in base $7$, $a 
eq 0$.  Also, because $N$ has three digits in base $12$, $c 
eq 0$.  Finally, because $N$ has three digits in both base $7$ and base $12$, we have $a,b,c < 7$. From the equation $7^2a + 7b + c = 12^2c + 12b + a$, we get $48a + 7b = 143c$. Since $c 
eq 0$, we must have $c \geq 1$.  Also note that $48a + 7b \leq 6\cdot48 + 6\cdot7 = 342 < 143\cdot 2$, so $c = 1$. Then the equation $48a + 7b = 143$ gives us $a = 3$ and $b = 5$. Therefore, the number $N$ is $3\cdot7^2 + 5\cdot 7 + 1 = \boxed{184}$ in base $10$.