5Frage: Wie viele der 100 kleinsten positiven ganzen Zahlen sind kongruent zu 3 (mod 7)?
["Title: How Many of the 100 Smallest Positive Integers Are Congruent to 3 (mod 7)?", "When exploring modular arithmetic, one common question is: How many numbers among the first 100 positive integers are congruent to 3 modulo 7? Understanding congruence in this context helps unlock patterns in number sequences and has practical applications in coding, cryptography, and problem-solving.", "In number theory, two integers ( a ) and ( b ) are said to be congruent modulo ( n ) if their difference is divisible by ( n ). That is, ( a \equiv b \pmod{n} ) if ( n \mid (a - b) ). In this article, we focus on how many of the numbers from 1 to 100 satisfy:", "[\nx \equiv 3 \pmod{7}\n]", "---", "### What Does It Mean for a Number to Be 3 (mod 7)?", "Being congruent to 3 mod 7 means that when divided by 7, the remainder is 3. So the numbers satisfying this condition take the form:", "[\nx = 7k + 3\n]", "where ( k ) is a non-negative integer (i.e., ( k = 0, 1, 2, \ldots )).", "---", "### Finding All Such Numbers ≤ 100", "We now determine how many values of ( k ) make ( 7k + 3 \leq 100 ).", "Start by solving the inequality:", "[\n7k + 3 \leq 100\n]", "Subtract 3 from both sides:", "[\n7k \leq 97\n]", "Divide by 7:", "[\nk \leq \frac{97}{7} \approx 13.857\n]", "Since ( k ) must be an integer, the largest possible value of ( k ) is 13.", "---", "### Listing the Valid Values", "With ( k = 0 ) to ( k = 13 ), we generate:", "- ( k = 0 ): ( 7(0) + 3 = 3 )\n- ( k = 1 ): ( 7(1) + 3 = 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 )", "These 14 numbers (3, 10, 17, ..., 94) are all ≤ 100.", "---", "### Conclusion: 14 Numbers Among the First 100 Are Congruent to 3 mod 7", "Out of the 100 smallest positive integers, exactly 14 are congruent to 3 modulo 7.", "This result demonstrates a predictable pattern in modular arithmetic — every 7th number starting from 3 satisfies this congruence. Recognizing such patterns helps efficiently solve number theory problems and has real-world uses, for example, in hashing algorithms or distributing workloads evenly.", "---", "Keywords:\npositive integers 1 to 100, congruent to 3 mod 7, modular arithmetic, math pattern, number theory, 7 moduli, mathematical pattern, 100 smallest numbers, arithmetic sequence, math education, problem solving in modular arithmetic", "Meta Description:\nDiscover how many of the first 100 positive integers satisfy ( x \equiv 3 \pmod{7} ). Learn the modular arithmetic behind this count and explore its applications."]
