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)?

Thu, 16 Apr 2026 15:36:49 +0000

Understanding Animal Migration Cycles Through Arithmetic Progressions: A Zoological Insight Animal migration is a remarkable natural phenomenon observed across species, from birds crossing continents to fish returning to their natal spawning grounds. Zoologists studying these patterns often find that certain migratory behaviors follow predictable, recurring sequences. In recent research, a zoologist noticed that some species return to specific regions at regular intervals—sometimes every 3 years, or more complex time frames resembling mathematical patterns, including arithmetic progressions. While migration cycles can vary in complexity, understanding the underlying periodicity helps scientists model movement and protect key habitats. Among the mathematical tools used in such studies, modular arithmetic—particularly congruences like n ≡ k (mod m)—plays a crucial role in identifying recurring patterns over time. 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?