4. Given is an integer $n\geq 1$. Find out the number of possible values of products $k \cdot m$, where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$.
To find the number of possible values of products \( k \cdot m \), where \( k \) and \( m \) are integers satisfying \( n^2 \leq k \leq m \leq (n+1)^2 \), we can use a lemma from the Indian Mathematical Olympiad (IMO) 1998. The lemma states that there do not exist four different positive integers \( a, b, c, d \) with \( n^2 \leq a, b, c, d < (n+1)^2 \) such that \( ad = bc \).

To prove this lemma, we assume the contrary and show a contradiction.

1. **Assume the contrary:**
   Suppose there exist four different positive integers \( a, b, c, d \) such that \( n^2 \leq a, b, c, d < (n+1)^2 \) and \( ad = bc \).

2. **Substitute the integers:**
   Let \( a = pq \), \( d = rs \), \( b = pr \), and \( c = qs \), where \( p, q, r, s \) are positive integers.

3. **Order the integers:**
   Assume \( a < b < c < d \). This implies \( c > a \) and \( b > a \).

4. **Derive inequalities:**
   Since \( c > a \), we have \( qs > pq \), which implies \( s \geq p + 1 \).
   Since \( b > a \), we have \( pr > pq \), which implies \( r \geq q + 1 \).

5. **Combine the inequalities:**
   Using the inequalities derived, we get:
   \[
   d = rs \geq (p+1)(q+1) = pq + p + q + 1
   \]
   Since \( pq \geq n^2 \), we have:
   \[
   d \geq n^2 + p + q + 1
   \]
   Using the arithmetic mean-geometric mean inequality (AM-GM inequality), we get:
   \[
   p + q \geq 2\sqrt{pq} \geq 2n
   \]
   Therefore:
   \[
   d \geq n^2 + 2n + 1 = (n+1)^2
   \]

6. **Contradiction:**
   The inequality \( d \geq (n+1)^2 \) contradicts the assumption that \( d < (n+1)^2 \). Hence, our initial assumption is false.

7. **Conclusion:**
   Since there do not exist four different positive integers \( a, b, c, d \) with \( n^2 \leq a, b, c, d < (n+1)^2 \) such that \( ad = bc \), the number of possible values of products \( k \cdot m \) is the number of distinct products formed by pairs \( (k, m) \) within the given range.

The final answer is \(\boxed{True}\).