Can a DPDA decide if two letters appear the same number of times mod 5?
What does $|w|_0 = |w|_1 \mod 5$ means ?
It could either mean (depending on your book's notations) :


$\exists k\in\mathbb Z, |w|_0 + 5k = |w|_1$
The remainder from the Euclidean Division of $|w|_1$ by 5 is $|w|_0$


Let's first resolve the first case. You should accept if and only if  5 divides $|w|_1 - |w|_0$. This can be done with a finite automaton with 5 states, to keep track of $|w|_1 - |w|_0\mod 5$ and only accepting when it is in state 0 (I'll let you find the transitions).

In the second case, you need to also ensure that the value of $|w|_0$ never goes beyond 4. That can be done by copying the same finite automaton as above 4 times, fixing the transitions accordingly (I would recommend to try doing that) and adding a "trash state" to go to if you have read more than four 0.

So we have deterministic finite state automata for both those languages. Finite state automata are just specific cases of PDA where you never use the stack. So there you have a DPDA for each of those languages.



Another way to do this, which is probably what the author was looking for, is to try to build a grammar. Hint : such a grammar has 5 non-terminal symbols, 2 terminals symbols, and 11 rules (this is for the first case). 
Those two constructions are equivalent though.