Thus, there are $ \boxed{14} $ integers between 1 and 100 that are congruent to 3 modulo 7.

["14 Integers Between 1 and 100 Congruent to 3 Modulo 7", "Understanding modular arithmetic is essential in number theory, and one fascinating problem is identifying integers within a range that satisfy a given congruence. In this article, we explore a key fact: there are exactly $ \boxed{14} $ integers between 1 and 100 that are congruent to 3 modulo 7. Let’s break down how this result is derived and why it matters.", "### What Does It Mean for an Integer to Be Congruent to 3 Modulo 7?", "An integer $ a $ is congruent to 3 modulo 7 if it satisfies:\n$$ a \equiv 3 \pmod{7} $$\nThis means when $ a $ is divided by 7, the remainder is 3. Equivalently, such integers can be written in the form:\n$$ a = 7k + 3 $$\nwhere $ k $ is any integer.", "### Finding All Valid Integers Between 1 and 100", "To find all integers between 1 and 100 that meet this condition, we substitute values of $ k $ into $ 7k + 3 $ and check which results lie in the desired range.", "Start with the smallest $ k $ such that $ 7k + 3 \geq 1 $:\n- $ k = 0 \Rightarrow 7(0) + 3 = 3 $ — valid\n- $ k = 1 \Rightarrow 7(1) + 3 = 10 $ — valid\n- Continue incrementing $ k $ until $ 7k + 3 > 100 $", "Now calculate the largest $ k $:\n$$ 7k + 3 \leq 100 $$\n$$ 7k \leq 97 $$\n$$ k \leq \frac{97}{7} \approx 13.857 $$\nSince $ k $ must be an integer, the maximum valid $ k $ is 13.", "Thus, $ k $ ranges from 0 to 13, inclusive. Counting these:\n$$ k = 0, 1, 2, \dots, 13 $$ — a total of $ 14 $ values.", "### Listing the 14 Integers", "Using $ a = 7k + 3 $ for $ k = 0 $ to $ 13 $:\n- $ k = 0 $: 3\n- $ k = 1 $: 10\n- $ k = 2 $: 17\n- $ k = 3 $: 24\n- $ k = 4 $: 31\n- $ k = 5 $: 38\n- $ k = 6 $: 45\n- $ k = 7 $: 52\n- $ k = 8 $: 59\n- $ k = 9 $: 66\n- $ k = 10 $: 73\n- $ k = 11 $: 80\n- $ k = 12 $: 87\n- $ k = 13 $: 94", "So the 14 integers are:\n3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94", "### Why This Matters", "This simple counting problem reveals important structure in modular arithmetic. Knowing how many integers within a range satisfy a given congruence helps in algorithm design, cryptography, coding theory, and optimization problems across computer science and mathematics.", "### Conclusion: A Clear Answer with Broad Implications", "Thus, there are exactly $ \boxed{14} $ integers between 1 and 100 that are congruent to 3 modulo 7. This result highlights how modular reasoning systematically solves counting problems and provides insights into the distribution of numbers across cycles—butterfly patterns modulo 7. Whether you're programming, studying discrete math, or just curious about number patterns, understanding such concepts enriches problem-solving skills and reveals hidden order in raw data.", "---\nKeywords for SEO: $ \boxed{14} $ integers, modulo 7 congruence, numbers between 1 and 100 congruent to 3, modular arithmetic examples, counting residues, math education, number theory concept, algorithmic counting, cryptography applications."]