What is the HCF of $3^5\times 5^4\times 7^2$ and $3^3\times 5^6\times 11$?

The HCF takes the LOWEST power of each prime that appears in BOTH numbers. Only 3 and 5 are shared, so the answer is $3^3\times 5^4=16875$; the $7^2$ belongs to the first number alone and the 11 to the second, so neither can appear. The trap $3^5\times 5^6$ takes the highest powers instead, which is the recipe for the LCM (although the LCM would also carry the $7^2$ and the 11). Adding the indices gives $3^8\times 5^{10}$, which is neither HCF nor LCM but the product's shape. And $3^3\times 5^4\times 7^2\times 11$ throws in every prime on show, ignoring the requirement that a common factor must divide both.