Question: A science educator is designing a modular arithmetic activity and asks: How many of the first 200 positive integers leave a remainder of 3 when divided by 7?
["Title: How Many of the First 200 Positive Integers Leave a Remainder of 3 When Divided by 7?", "When introducing students to modular arithmetic, one engaging and classic activity involves identifying numbers that satisfy specific remainders upon division. A common and insightful question is: How many of the first 200 positive integers leave a remainder of 3 when divided by 7? This question helps learners explore modular congruences while applying practical number theory concepts.", "### Understanding Modular Arithmetic Remainders", "In modular arithmetic, the expression “a leaves a remainder of b when divided by m” is mathematically expressed as:\n[ a \equiv b \pmod{m} ]\nThis means that when a is divided by m, the remainder is b. For this problem, we want:\n[ n \equiv 3 \pmod{7} ]", "### Finding Numbers That Satisfy the Condition", "Numbers that satisfy ( n \equiv 3 \pmod{7} ) can be written in the form:\n[\nn = 7k + 3\n]\nwhere k is a non-negative integer. Our goal is to find all such n that satisfy:\n[ 1 \leq n \leq 200 ]", "### Solving the Inequality", "Substitute ( n = 7k + 3 ) into the inequality:\n[\n1 \leq 7k + 3 \leq 200\n]", "Subtract 3 from all parts:\n[\n-2 \leq 7k \leq 197\n]", "Since k must be a non-negative integer (because n is positive and starts at 3), the smallest value of k is 0. Now solve for the largest integer k:\n[\n7k \leq 197 \quad \Rightarrow \quad k \leq \frac{197}{7} \approx 28.14\n]\nThus, the largest whole number k is 28.", "### Counting Valid Values of k", "k ranges from 0 to 28, inclusive. This gives:\n[\nk = 0, 1, 2, \dots, 28\n]\nThere are ( 28 - 0 + 1 = 29 ) values.", "### List of Examples and Verification", "- For k = 0: ( n = 7(0) + 3 = 3 )\n- For k = 28: ( n = 7(28) + 3 = 196 + 3 = 199 ), which is ≤ 200", "All these values are within the desired range and yield a remainder of 3 when divided by 7.", "### Conclusion", "There are exactly 29 positive integers among the first 200 that leave a remainder of 3 upon division by 7. This activity helps students recognize patterns in modular systems, apply algebraic expressions to number sequences, and deepen their understanding of division with remainders.", "---", "Teaching Tip:** Use a simple table or sequence exploration to show how the numbers increment by 7:\n3, 10, 17, 24, ..., 199 — this visual approach reinforces the arithmetic progression and strengthens numerical intuition.", "---", "Mastering such problems lays a strong foundation for more advanced topics in number theory, cryptography, and computer science — making this seemingly simple question a gateway to richer mathematical exploration."]
