Minimum

I presume that we are working in real number system ! So Minimum can be $-\infty$ . because it is not necessary that a,b,c,..........z are positive real's . (So option $26\sqrt [ 26 ]{ 100 }$ . is rejected )

without loss of generality Let $a\quad =\lim _{ a\rightarrow -\infty }{ (a) }$ . and also Let $b\quad =\quad \lim _{ b\rightarrow 0 }{ (b) }$ . and again let al other alphabhet's (Real number's ) are aribitary number's which has finite representation in Real number system ( say 2 ,-8 , +11 , -19, $\sqrt { 2 } ,\sqrt [ 3 ]{ 1729 }$ . ....etc.)

So by fundamental rule of limit $\quad \lim _{ a\rightarrow -\infty }{ (a) } \times \lim _{ b\rightarrow 0 }{ (b) } \quad =\quad L\quad (can\quad be\quad exist)$ .

And According to question It must has to exist (since $a*b*c*d........z=100$ . )

Therefore ${ (a+b+c+\quad .\quad .\quad .\quad .\quad +z) }_{ min }\quad \longrightarrow \quad -\infty$ .

If all letters except for $z$ are very large negative numbers, then $z$ has to be a very small negative number in order for all letters to multiply to 100. (E.g. consider $-b$ where $b$ is very large). This means that the minimum value consists of 25 negative big numbers and one negative small number. This adds to a very big negative number. Hence, the larger all letters (but negative and large such as $-10000000000$ except for $z$ , the smaller the minimum and hence it tends to $-\infty$ as you can make all letters except for $z$ negatively larger and larger.