Realization of divide by 78 counter
The book is wrong. The right answer is C. 78=13*6. The mod-13 counter will be incremented with each pulse of the incoming signal, and will overflow every 13 counts of such pulses. The mod-6 counter will be incremented each time that the mod-13 counter overflows. The mod-6 counter will overflow every 6 of its counts. Therefore, the mod-6 counter will overflow every 6*13=78 pulses of the incoming signal.

In other words, for each state of the mod-6 counter, the mod-13 counter counts 13 states. So, in total, you will have 6*13=78 different "global" states. That's why, if you have them one after the other (so that counter k increments one unit when counter k-1 overflows), you have to multiply the numbers of states, not add them.

These will be the count values for each of the two counters, before each pulse at the input.

{(mod-13 counter value) (mod-6 counter value)}  
{0 0}  
{1 0}  
...  
{12 0}  
{0 1}  
{1 1}  
...  
{12 1}  
{0 2}  
{1 2}  
...  
...  
{12 5}  
&lt;-- Here is the overflow for the global counter formed. Above this line, there are 78 different states.  
{0 0}  
...  


Also:
a) and d) are wrong, because there is no integer N such that \$13^N=78\$.
b) is wrong, because there is no integer N such that \$6^N=78\$.