Let a common tangent to the curves $${y^2} = 4x$$ and $${(x - 4)^2} + {y^2} = 16$$ touch the curves at the points P and Q. Then $${(PQ)^2}$$ is equal to __________.

The sum of all those terms, of the arithmetic progression 3, 8, 13, ...., 373, which are not divisible by 3, is equal to ____________.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons, who participated in the tournament, is ___________.

Let a, b, c be three distinct positive real numbers such that $${(2a)^{{{\log }_e}a}} = {(bc)^{{{\log }_e}b}}$$ and $${b^{{{\log }_e}2}} = {a^{{{\log }_e}c}}$$.

Then, 6a + 5bc is equal to ___________.