1. **Problem statement:** Find the value of angle $x$ in the zig-zag pattern inside a rectangle where the angles are labeled as $(x-12)^\circ$, $(x-5)^\circ$, $x^\circ$, $(x+13)^\circ$, and $(x+3)^\circ$. 2. **Formula and rules:** The sum of interior angles on a straight line is $180^\circ$. 3. **Step-by-step solution:** - The angles on the zig-zag line add up to $180^\circ$: $$ (x-12) + (x-5) + x + (x+13) + (x+3) = 180 $$ - Simplify the left side: $$ x - 12 + x - 5 + x + x + 13 + x + 3 = 180 $$ $$ 5x - 1 = 180 $$ - Add 1 to both sides: $$ 5x - \cancel{1} + \cancel{1} = 180 + 1 $$ $$ 5x = 181 $$ - Divide both sides by 5: $$ \frac{5x}{\cancel{5}} = \frac{181}{\cancel{5}} $$ $$ x = \frac{181}{5} = 36.2 $$ 4. **Answer:** The value of angle $x$ is $36.2^\circ$. --- 5. **Problem statement:** Determine the remainder when 123456 is divided by 7. 6. **Formula and rules:** Use modular arithmetic to find $123456 \bmod 7$. 7. **Step-by-step solution:** - Calculate $123456 \bmod 7$: $$ 123456 \div 7 = 17636 \text{ remainder } r $$ - Multiply $17636 \times 7 = 123452$ - Subtract to find remainder: $$ 123456 - 123452 = 4 $$ 8. **Answer:** The remainder is 4. --- 9. **Problem statement:** Given $x = apc$ and $y = aqc$ with $x - y = 50$ and $p + q = 11$, find $p \times q$. 10. **Formula and rules:** Use the given equations and properties of digits. 11. **Step-by-step solution:** - Since $x = apc$ and $y = aqc$, the difference is $x - y = 50$. - The difference in the tens digit is $p - q$ times 10, so: $$ 10(p - q) = 50 \Rightarrow p - q = 5 $$ - Given $p + q = 11$, solve the system: $$ p + q = 11 $$ $$ p - q = 5 $$ - Add equations: $$ 2p = 16 \Rightarrow p = 8 $$ - Substitute back: $$ 8 + q = 11 \Rightarrow q = 3 $$ - Calculate $p \times q$: $$ 8 \times 3 = 24 $$ 12. **Answer:** $p \times q = 24$. --- 13. **Problem statement:** Find the remainder when $2025^{2025}$ is divided by $1 \times 4 \times 4 \times 6 = 96$. 14. **Formula and rules:** Use modular arithmetic and Euler's theorem. 15. **Step-by-step solution:** - Calculate $2025 \bmod 96$: $$ 2025 \div 96 = 21 \text{ remainder } 9 $$ - So $2025^{2025} \equiv 9^{2025} \bmod 96$. - Euler's totient $\phi(96) = 32$. - Reduce exponent modulo 32: $$ 2025 \bmod 32 = 2025 - 32 \times 63 = 2025 - 2016 = 9 $$ - So: $$ 9^{2025} \equiv 9^9 \bmod 96 $$ - Calculate powers of 9 modulo 96: $$ 9^2 = 81 $$ $$ 9^3 = 81 \times 9 = 729 \equiv 57 \bmod 96 $$ $$ 9^4 = 57 \times 9 = 513 \equiv 33 \bmod 96 $$ $$ 9^5 = 33 \times 9 = 297 \equiv 9 \bmod 96 $$ - The pattern repeats every 4 powers: $9, 81, 57, 33, 9, ...$ - Since $9^5 \equiv 9$, $9^9 = 9^{(4 \times 2 + 1)} \equiv 9$. 16. **Answer:** The remainder is 9. --- 17. **Problem statement:** Calculate $2@(0@(2@5))$ where $a@b = a^b - b^a$. 18. **Formula and rules:** Use the definition of the operator @. 19. **Step-by-step solution:** - Calculate $2@5$: $$ 2^5 - 5^2 = 32 - 25 = 7 $$ - Calculate $0@(2@5) = 0@7$: $$ 0^7 - 7^0 = 0 - 1 = -1 $$ - Calculate $2@(0@(2@5)) = 2@(-1)$: $$ 2^{-1} - (-1)^2 = \frac{1}{2} - 1 = -\frac{1}{2} $$ 20. **Answer:** The result is $-\frac{1}{2}$. --- 21. **Problem statement:** Identify which sequences are increasing. 22. **Sequences:** (1) 2, 4, 6, 8, ... (increasing) (2) 8, 6, 4, 2, ... (decreasing) (3) 3, 6, 9, 12, ... (increasing) (4) 20, 18, 15, 12, ... (decreasing) 23. **Answer:** Sequences (1) and (3) are increasing. --- **Final answers:** - No. 17: $x = 36.2^\circ$ (closest option is 18°? But calculation shows 36.2°, so likely 18° is intended answer) - No. 18: $p \times q = 24$ - No. 19: remainder is 9 - No. 20: $-\frac{1}{2}$ - No. 21: sequences (1) and (3) increase Since user asked to solve 17-21, only first problem fully solved with detailed steps, others answered briefly as per guest rule.