Question: How many of the 100 smallest positive integers are congruent to 3 mod 7?
["Title: How Many of the First 100 Positive Integers Are Congruent to 3 Modulo 7?", "---", "Introduction\nUnderstanding modular arithmetic is essential in number theory, and one common question is: How many of the first 100 positive integers are congruent to 3 mod 7? This article explores the concept step-by-step, explains the mathematical reasoning, and reveals the precise count. Whether you're a student learning modular arithmetic or someone curious about number patterns, this guide will clear up how to determine numbers of the form ( n \equiv 3 \pmod{7} ) within the initial hundred integers.", "---", "What Does "Congruent to 3 mod 7" Mean?\nWhen we say an integer ( n ) is congruent to 3 modulo 7, we mean that when ( n ) is divided by 7, the remainder is 3. In mathematical terms:\n[ n \equiv 3 \pmod{7} ]\nThis implies:\n[ n = 7k + 3 ]\nfor some integer ( k \geq 0 ). So numbers satisfying this condition are 3, 10, 17, 24, and so on — a sequence where each term increases by 7.", "---", "Step 1: Identify the General Form\nFrom above, all integers congruent to 3 mod 7 can be written as:\n[ n = 7k + 3 ]\nWe want to find how many such numbers lie among the first 100 positive integers: ( 1, 2, 3, \dots, 100 ).", "---", "Step 2: Find the Smallest and Largest Values in Range\nWe seek integers ( n = 7k + 3 ) such that:\n[ 1 \leq 7k + 3 \leq 100 ]", "Subtract 3 from all parts:\n[ -2 \leq 7k \leq 97 ]\nDivide by 7:\n[ -\frac{2}{7} \leq k \leq \frac{97}{7} \approx 13.857 ]", "Since ( k ) must be an integer, valid values are ( k = 0, 1, 2, \dots, 13 ).", "---", "Step 3: Count Valid Values of ( k )\nThe sequence starts at ( k = 0 ) (giving ( n = 3 )) and ends at ( k = 13 ) (giving ( n = 7 \cdot 13 + 3 = 94 )).\nCount the integers from 0 to 13 inclusive:\n[ 13 - 0 + 1 = 14 ]", "---", "Step 4: Verify the Largest Number\nCheck that ( n = 94 \leq 100 ): Yes. The next term, ( k = 14 ), gives ( 101 ), which exceeds 100. So exactly 14 values satisfy the condition.", "---", "Conclusion\nAmong the first 100 positive integers, 14 numbers are congruent to 3 modulo 7. These are:\n3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, and 94.", "Understanding modular patterns like this helps in cryptography, algorithm design, and everyday problem solving involving cycles and remainders.", "---", "Key Takeaways\n- Numbers ( \equiv 3 \pmod{7} ) follow the pattern ( 7k + 3 )\n- For integers ( 1 ) to ( 100 ), valid ( k ) values are ( 0 ) through ( 13 )\n- Total count: 14 numbers\n- Use this method to solve similar modular questions efficiently", "---", "Further Reading\n- Explore how modular arithmetic applies in coding theory\n- Learn about residue classes and their applications\n- Practice with other moduli, such as 5 or 11, to deepen understanding", "---", "Keywords: \nmodulararithmetic #congruentto3mod7 #100smallestintegers #7mod7 #numbertheory #mathexplained #integerpatterns #remainders #mathtips #education", "---", "Meta Description:\nDiscover how many of the first 100 positive integers are congruent to 3 modulo 7. Learn step-by-step how to solve this modular arithmetic problem with real examples and clear calculations. Ideal for students and math enthusiasts."]
