Solution: Numbers congruent to 3 mod 7 are $3, 10, 17, \ldots, 3 + 7k$. The largest such number $\leq 100$ satisfies $3 + 7k \leq 100 \Rightarrow k \leq rac{97}{7} pprox 13.857$. Thus, $k = 0, 1, \ldots, 13$, giving 14 numbers. The count is $oxed{14}$.

["Solution Explained: How Numbers Congruent to 3 Mod 7 Relate to the Range Up to 100", "Understanding the pattern of numbers congruent to 3 modulo 7 offers valuable insight into modular arithmetic and sequences in number theory. The key idea is that any number congruent to 3 mod 7 can be expressed in the form:", "$$\n3 + 7k\n$$", "where $k$ is a non-negative integer. This formula generates a carefully structured sequence: $3, 10, 17, 24, \ldots$, continuing by adding 7 each time.", "To find how many such numbers exist that are less than or equal to 100, solve the inequality:", "$$\n3 + 7k \leq 100\n$$", "Subtracting 3 from both sides gives:", "$$\n7k \leq 97\n$$", "Dividing by 7 yields:", "$$\nk \leq \frac{97}{7} \approx 13.857\n$$", "Since $k$ must be an integer, the largest possible value is $k = 13$. Including $k = 0$, which gives the first number $3$, the valid values of $k$ range from $0$ to $13$ inclusive.", "This gives a total of:", "$$\n13 - 0 + 1 = 14 \ ext{ numbers}\n$$", "Thus, there are $\boxed{14}$ positive integers less than or equal to 100 that are congruent to 3 modulo 7.", "This solution illustrates the elegance of arithmetic sequences and modular reasoning, empowering students and learners to recognize patterns and compute solution counts efficiently."]