Let's solve the expression step by step:

Expression:

$(32^3 \mod 13)^2 \mod 11$

Step 1: Simplify $32^3 \mod 13$

1. Find $32 \mod 13$: $32 \div 13 = 2 \text{ remainder } 6$ Thus, $32 \equiv 6 \mod 13$.

2. Calculate $6^3 \mod 13$: $6^3 = 6 \times 6 \times 6 = 216$ Now find $216 \mod 13$: $216 \div 13 = 16 \text{ remainder } 8$ Therefore, $6^3 \equiv 8 \mod 13$.

Step 2: Square the result and find modulo 11

Now we need to calculate: $8^2 \mod 11$

1. Calculate $8^2$: $8^2 = 64$

2. Find $64 \mod 11$: $64 \div 11 = 5 \text{ remainder } 9$ Thus, $64 \equiv 9 \mod 11$.

Final Result:

$(32^3 \mod 13)^2 \mod 11 = 9$

So, the final answer is 9.

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Do you want more details, or have any questions? Here are five related problems to try:

1. Calculate $(27^3 \mod 17)^2 \mod 13$.
2. Find $(15^4 \mod 19)^3 \mod 7$.
3. Simplify $(45^2 \mod 23)^2 \mod 5$.
4. Evaluate $(8^5 \mod 14)^3 \mod 9$.
5. Solve $(100^3 \mod 31)^2 \mod 16$.

Tip: When dealing with large exponents in modular arithmetic, simplify the base first before raising it to the power.