Express $\frac{165_7}{11_2}+\frac{121_6}{21_3}$ in base 10.
We begin by converting all the numbers to base 10: \begin{align*} 165_7&=1(7^2)+6(7^1)+5(7^0)=49+42+5=96\\
11_2&=1(2^1)+1(2^0)=2+1=3\\
121_6&=1(6^2)+2(6^1)+1(6^0)=36+12+1=49\\
21_3&=2(3^1)+1(3^0)=6+1=7
\end{align*}Thus, the original expression becomes $\frac{96}{3}+\frac{49}{7}=32+7=\boxed{39}$.
39
Level 3
Number Theory
Problem: Express $\frac{165_7}{11_2}+\frac{121_6}{21_3}$ in base 10.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the final answer, we first need to convert each number from its respective base to base 10. 

Starting with the first fraction, $\frac{165_7}{11_2}$:
1. Convert $165_7$ to base 10:
   - The digit '1' in the 7^2 place contributes $1 \times 7^2 = 49$.
   - The digit '6' in the 7^1 place contributes $6 \times 7^1 = 42$.
   - The digit '5' in the 7^0 place contributes $5 \times 7^0 = 5$.
   - Summing these values gives $49 + 42 + 5 = 96$.
   - Thus, $165_7 = 96_{10}$.

2. Convert $11_2$ to base 10:
   - The digit '1' in the 2^1 place contributes $1 \times 2^1 = 2$.
   - The digit '1' in the 2^0 place contributes $1 \times 2^0 = 1$.
   - Summing these values gives $2 + 1 = 3$.
   - Thus, $11_2 = 3_{10}$.

3. Now compute the first fraction:
   - $\frac{165_7}{11_2} = \frac{96}{3} = 32$.

Next, we move to the second fraction, $\frac{121_6}{21_3}$:
1. Convert $121_6$ to base 10:
   - The digit '1' in the 6^2 place contributes $1 \times 6^2 = 36$.
   - The digit '2' in the 6^1 place contributes $2 \times 6^1 = 12$.
   - The digit '1' in the 6^0 place contributes $1 \times 6^0 = 1$.
   - Summing these values gives $36 + 12 + 1 = 49$.
   - Thus, $121_6 = 49_{10}$.

2. Convert $21_3$ to base 10:
   - The digit '2' in the 3^1 place contributes $2 \times 3^1 = 6$.
   - The digit '1' in the 3^0 place contributes $1 \times 3^0 = 1$.
   - Summing these values gives $6 + 1 = 7$.
   - Thus, $21_3 = 7_{10}$.

3. Now compute the second fraction:
   - $\frac{121_6}{21_3} = \frac{49}{7} = 7$.

Finally, we combine the results from both fractions:
- $32 + 7 = 39$.

Thus, the final answer is $39$.<|END_OF_LATENT|> The final answer is $39$.