Question: How many of the first 150 positive integers are congruent to 2 modulo 7?

Mon, 04 May 2026 08:18:42 +0000

**How Many of the First 150 Positive Integers Are Congruent to 2 Modulo 7?** When curious minds ask how many of the first 150 positive integers are congruent to 2 modulo 7, a quiet mathematical pattern emerges—one that reflects deeper rules governing numbers and patterns in everyday life. The question isn’t just academic; it’s part of a growing trend in the US where people explore number theory for logic, coding, and pattern recognition. With mobile-first curiosity setting the pace, this query reflects a desire to understand structure behind sequences, a mindset fueling learning about everything from app algorithms to financial modeling. **Understanding the Pattern: Congruent to 2 Modulo 7** Numbers congruent to 2 mod 7 follow a repeating cycle: 2, 9, 16, 23, ..., each separated by 7. This sequence continues modulo 7 because every 7th step wraps around the remainder. To find how many such numbers exist below 151, divide 150 by 7. The result is approximately 21.4, meaning 21 full cycles fit within 150. The largest number in this series under 151 is 148 — confirmed by calculating 7×21 + 2 = 149, but wait — 7×21 = 147 → 147 + 2 = 149. That places 149 as the final term, so the count is exactly 21. The next one would be 156, beyond the limit.