Question:** A zoologist studying animal migration patterns observes that certain species return every few years, forming a sequence similar to an arithmetic progression. How many of the first 50 positive integers are congruent to 3 (mod 7)? - Baxtercollege

Understanding the Context

How Many of the First 50 Positive Integers Are Congruent to 3 (mod 7)?
To explore how mathematical patterns appear in nature, consider this key question: How many of the first 50 positive integers are congruent to 3 modulo 7?

Two integers are congruent modulo 7 if they differ by a multiple of 7. That is, a number n satisfies:
n ≡ 3 (mod 7)
if when divided by 7, the remainder is 3. These numbers form an arithmetic sequence starting at 3 with a common difference of 7:
3, 10, 17, 24, 31, 38, 45

This is the sequence of positive integers congruent to 3 mod 7, within the first 50 integers.

To count how many such numbers exist, we solve:
Find all integers n such that:
3 ≤ n ≤ 50
and
n ≡ 3 (mod 7)

Key Insights

We can express such numbers as:
n = 7k + 3
Now determine values of k for which this remains ≤ 50.

Solve:
7k + 3 ≤ 50
7k ≤ 47
k ≤ 47/7 ≈ 6.71

Since k must be a non-negative integer, possible values are k = 0, 1, 2, 3, 4, 5, 6 — a total of 7 values.

Thus, there are 7 numbers among the first 50 positive integers that are congruent to 3 modulo 7.

Linking Zoology and Math
Just as migration cycles may follow periodic patterns modeled by modular arithmetic, zoologists continue to uncover deep connections between nature’s rhythms and mathematical structures. Identifying how many numbers in a range satisfy a given congruence helps quantify and predict biological phenomena—key for conservation and understanding species behavior.

Final Thoughts

This intersection of ecology and mathematics enriches our appreciation of wildlife cycles and underscores how number theory can illuminate the natural world. Whether tracking bird migrations or analyzing habitat use, recurring sequences like those defined by n ≡ 3 (mod 7) reveal nature’s elegant order.

Conclusion
Using modular arithmetic, researchers efficiently identify recurring patterns in animal migration. The fact that 7 of the first 50 positive integers are congruent to 3 mod 7 illustrates how simple mathematical rules can describe complex biological timing. A zoologist’s observation becomes a bridge between disciplines—proving that behind every migration lies not just instinct, but also an underlying mathematical harmony.