Find the HCF of $2^5\times 3^4\times 7^2$ and $2^3\times 3^6\times 11^3$.

For the HCF take the LOWEST power of every prime that appears in BOTH numbers. Only 2 and 3 are common: 2 to the power min(5, 3) = $2^3$, and 3 to the power min(4, 6) = $3^4$. So HCF = $2^3\times 3^4$. Taking the highest powers instead gives the LCM part $2^5\times 3^6$, and including $7^2$ or $11^3$ is wrong because neither prime occurs in both numbers.