A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.
1. **Define the set \( S \):**
   We start by considering the set \( S = \{2^{a_1}3^{a_2}5^{a_3}7^{a_4} \mid a_1, a_2, a_3, a_4 \geq 0\} \). This set \( S \) contains all integers whose prime factors are less than 10, specifically the primes 2, 3, 5, and 7.

2. **Representation of elements in \( S \):**
   Each element in \( S \) can be uniquely represented by a quadruplet \((a_1, a_2, a_3, a_4)\), where \( a_i \) are non-negative integers representing the exponents of the primes 2, 3, 5, and 7 respectively.

3. **Modulo 2 representation:**
   To determine when the product \( ab \) is a perfect square, we consider the exponents modulo 2. Specifically, we look at the quadruplets \((a_1, a_2, a_3, a_4) \mod 2\). Each \( a_i \) can be either 0 or 1 modulo 2, giving us \( 2^4 = 16 \) possible distinct representations.

4. **Pigeonhole Principle:**
   By the Pigeonhole Principle, if we have more than 16 elements in \( A \), there must be at least two elements, say \( a \) and \( b \), such that their corresponding quadruplets are congruent modulo 2. That is, \((a_1, a_2, a_3, a_4) \equiv (b_1, b_2, b_3, b_4) \pmod{2}\).

5. **Perfect square condition:**
   If \((a_1, a_2, a_3, a_4) \equiv (b_1, b_2, b_3, b_4) \pmod{2}\), then the product \( ab \) can be written as:
   \[
   ab = 2^{a_1 + b_1} 3^{a_2 + b_2} 5^{a_3 + b_3} 7^{a_4 + b_4}
   \]
   Since \( a_i \equiv b_i \pmod{2} \), each \( a_i + b_i \) is even, making \( ab \) a perfect square.

6. **Constructing \( A \) with 16 elements:**
   To show that 16 elements are not sufficient, we can construct \( A \) using all possible combinations of \( a_i \) being 0 or 1. This gives us exactly 16 distinct elements, each with a unique representation modulo 2. Therefore, no two elements in this set will have the same representation modulo 2, and thus no pair \( (a, b) \) will satisfy the condition that \( ab \) is a perfect square.

7. **Conclusion:**
   Therefore, the smallest \( n \) such that \( A \) must contain at least two distinct elements \( a \) and \( b \) such that \( ab \) is the square of an integer is \( n = 17 \).

The final answer is \( \boxed{17} \)