<s>[INST] A number N in base 10, is 503 in base b and 305 in base b + 2. What is the product of the digits of N? Pls help me with A number N in base 10, is 503 in base b and 305 in base b + 2. What is the product of the digits of N? I amen't solve it. Help needed
 [/INST] A number $\overline{xyz}$ in base $b$ is equal to $x\cdot b^2 + y\cdot b + z$ and so on for larger numbers continuing the pattern increasing the power of $b$ as we continue to the left.  For instance, in base 10 we have $35013$ is equal to $3\cdot 10^4 + 5\cdot 10^3 + 0\cdot 10^2 + 1\cdot 10 + 3$.
So, we are told that $N = 5\cdot b^2 + 0\cdot b + 3$ and also that $N = 3\cdot (b+2)^2 + 0\cdot (b+2) + 5$
Setting these equal, we can find $b$.
Now, armed with the knowledge of what $b$ is we can find $N$.
Now, armed with the knowledge of what $N$ is in decimal, we can find the product of the digits of $N$.
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