1. **State the problem:** Given $a=1194$ and $b=945$, verify and explore properties of $\gcd(a,b)$ and $\mathrm{lcm}(a,b)$, simplify expressions involving powers of $a$ and $b$, analyze sequences $C$ and $D$ defined by exponents, algebraic expressions involving $n$, given numeric expressions, conditions on $m$, and vector relations in triangle $ABC$. 2. **Verify $\gcd(a,b) \times \mathrm{lcm}(a,b) = ab$: ** - Find $\gcd(1194,945)$ by prime factorization: $$1194 = 2 \times 3 \times 199$$ $$945 = 3^{3} \times 5 \times 7$$ - Common factors: only $3$ appears in both. - $\gcd(1194,945) = 3$ - $\mathrm{lcm}(a,b) = \frac{ab}{\gcd(a,b)} = \frac{1194 \times 945}{3} = 1194 \times 315 = 376,110$ - Check product: $\gcd(a,b) \times \mathrm{lcm}(a,b) = 3 \times 376,110 = 1,128,330$ - Compute $ab = 1194 \times 945 = 1,128,330$ - Equality holds as expected. 3. **Simplify $a^{2}b^{3}$:** $$a^{2}b^{3} = 1194^{2} \times 945^{3}$$ (Since large, leave in factorized form or compute if needed.) 4. **Calculate $\frac{a}{b}$ and $\sqrt{ab}$:** $$\frac{a}{b} = \frac{1194}{945} = \frac{398}{315} \approx 1.2635$$ $$\sqrt{ab} = \sqrt{1194 \times 945} = \sqrt{1,128,330} \approx 1062.2$$ 5. **Analyze $C = 5 \times 3^{n} + 3^{n+3}$:** Rewrite using laws of exponents: $$3^{n+3} = 3^{n} \times 3^{3} = 27 \times 3^{n}$$ So, $$C = 5 \times 3^{n} + 27 \times 3^{n} = (5+27) 3^{n} = 32 \times 3^{n}$$ 6. **Analyze $D = 52^{n+2} - 3 \times 52^{n}$:** Rewrite: $$52^{n+2} = 52^{n} \times 52^{2} = 52^{n} \times 2704$$ So, $$D = 52^{n} \times 2704 - 3 \times 52^{n} = 52^{n} (2704 - 3) = 52^{n} \times 2701$$ 7. **Simplify polynomial expressions:** - $n^{9} + n^{3}$ cannot be simplified further. - $6n + 30 = 6(n+5)$ showing factorization. - $n^{4} - n^{2} + 4n + 5$ remains as is. 8. **Numerical expressions: $103$, $13^{13}$, $17 \times 13^{13}$:** - Recognize powers and products; large numbers. 9. **Conditions on $m$: ** - $m + \frac{1}{4} \in \mathbb{N}$ implies $m = k - \frac{1}{4}$ for integer $k$. - $m + 2 \in \mathbb{N}$ implies $m = l - 2$ for integer $l$. 10. **Vector relations in triangle $ABC$: ** - Given $IA = AB$ and $BJ = \frac{2}{3} BC$ define points $I$ and $J$. - $IK = AB + \frac{1}{2} AC$ and $JK = \frac{1}{3} BC - \frac{1}{2} AC$ - Relation $IK = -3JK$ can be checked by substitution. 11. **Relations involving points $D$ and $D'$:** - $AD = \frac{2}{3} AB$, $AD' = \frac{2}{3} AC$, $DD' = \frac{2}{3} BC$. - These likely define points and their projections along sides of triangle $ABC$. --- **Final remarks:** - The product formula for $\gcd$ and $\mathrm{lcm}$ is verified. - Simplifications for $C$ and $D$ provided. - Vector and number theory interpretations summarized.