Two positive integers m and n satisfy HCF (m, n) = 14 and LCM (m, n) = 168. If m < n and m divides 84, what is the maximum possible value of m?

Product of numbers =$$\operatorname{LCM}\times\ \operatorname{HCF}$$ = $$168\times\ 14$$

Given, $$m\ <\ n$$ and m divides 84. 

So, let us check the factors of 84 in descending order.

1. m = 84
   n = $$\ \frac{\ 168\times\ 14}{84}=28$$
   But this violates the condition of $$m\ <\ n$$ and cannot be the answer.
2. m = 42
   n = $$\ \frac{\ 168\times\ 14}{42}=56$$
   This also satisfies the other condition of $$m\ <\ n$$
   So, the maximum possible value of m is 42.