Two numbers are $a=2^3\times 3^{m}x$ 5 and $b=2^{n}x{3}^2\times 5^2$. Their HCF is $2^2\times 3^2\times$ 5 and their LCM is $2^3\times 3^4\times 5^2$.
What is the value of $m+n$?

Compare the exponents prime by prime. For 2: the HCF needs min(3, n) = 2, so n = 2, and then max(3, 2) = 3 matches the LCM. For 3: the LCM needs max(m, 2) = 4, so m = 4, and then min(4, 2) = 2 matches the HCF. Hence m + n = 4 + 2 = 6. Reading the HCF exponent as m gives 2 + 2 = 4; reading the LCM exponent as n gives 4 + 4 = 8.