Let $p$ and $q$ be relatively prime positive integers. A subset $S\subseteq \mathbb{N}_0$ is called ideal if $0 \in S$ and, for each element $n \in S$, the integers $n+p$ and $n+q$ belong to $S$. Determine the number of ideal subsets of $\mathbb{N}_0$.
1. **Understanding the Problem:**
   We need to determine the number of ideal subsets \( S \subseteq \mathbb{N}_0 \) where \( S \) contains 0 and for each \( n \in S \), the integers \( n + p \) and \( n + q \) also belong to \( S \). Here, \( p \) and \( q \) are relatively prime positive integers.

2. **Representation of Integers:**
   Since \( p \) and \( q \) are relatively prime, any integer \( x \) can be represented in the form \( x = ap + bq \) where \( a \) and \( b \) are integer coefficients. This is a consequence of the fact that \( p \) and \( q \) being relatively prime implies that their linear combinations can generate any integer.

3. **Ideal Subset Property:**
   Every ideal subset \( S \) must contain all integers of the form \( up + vq \) for non-negative integers \( u \) and \( v \). This is because if \( n \in S \), then \( n + p \in S \) and \( n + q \in S \), and by induction, all such combinations must be in \( S \).

4. **Characterizing Non-Members:**
   If an integer \( x \) is not in \( S \), it must be of the form \( x = ap + bq \) where either \( a \) or \( b \) is negative. We can reduce \( x \) to a form where \( -q < a < 0 \) and \( 0 < b < p \) with the condition \( \frac{b}{-a} > \frac{p}{q} \).

5. **Defining Ideal Subsets:**
   Every ideal subset is uniquely defined by a pair of sequences:
   \[
   q > -a_1 > -a_2 > \dots > -a_m > 0 \quad \text{and} \quad p > b_1 > b_2 > \dots > b_m > 0
   \]
   with the condition \( \frac{b_j}{-a_j} > \frac{p}{q} \) for all \( j = 1, \dots, m \).

6. **Counting Ideal Subsets:**
   The number of such sequences corresponds to the number of paths from \( (0,0) \) to \( (q,p) \) in the integer grid \([0,q] \times [0,p]\) where each step increments both coordinates and does not go below the line connecting the points \( (0,0) \) and \( (q,p) \).

7. **Lattice Path Counting:**
   The number of such paths is given by the binomial coefficient \( \binom{p+q}{p} \). This is because we are choosing \( p \) steps in one direction (say, horizontal) out of \( p+q \) total steps.

The final answer is \( \boxed{\binom{p+q}{p}} \).