Question: How many of the 100 smallest positive integers are congruent to $3 \pmod{7}$?
["How Many of the 100 Smallest Positive Integers Are Congruent to $3 \pmod{7}$?", "When exploring modular arithmetic, one frequently asked question is: How many numbers in the first 100 positive integers are congruent to $3$ modulo $7$? Understanding this concept helps unlock deeper insights into patterns in number sequences and is valuable in fields like cryptography, computer science, and number theory.", "### What Does “Congruent to 3 mod 7” Mean?", "An integer ( n ) is congruent to $3 \pmod{7}$ if when divided by 7, the remainder is 3. In mathematical terms:", "[\nn \equiv 3 \pmod{7} \quad \ ext{means} \quad n = 7k + 3 \quad \ ext{for some integer } k \geq 0\n]", "### Finding Numbers in the First 100 Positive Integers", "We want to count all such ( n ) satisfying:", "[\n1 \leq n \leq 100\n]\n[\nn = 7k + 3\n]", "Let’s solve for ( k ):", "[\n7k + 3 \leq 100\n]\n[\n7k \leq 97\n]\n[\nk \leq \frac{97}{7} \approx 13.857\n]", "Since ( k ) must be a non-negative integer, the largest possible ( k ) is 13.", "Now find the smallest ( k ):\nWhen ( k = 0 ), ( n = 3 ), which is within 1 to 100.", "So ( k ) ranges from 0 to 13 inclusive — that’s $ 14 $ values.", "### Verifying the Sequence", "List the numbers:", "For ( k = 0 ) to ( 13 ):", "[\nn = 3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94, 101\n]", "But 101 exceeds 100, so the valid values are:", "[\n3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94\n]", "There are 14 numbers in total.", "### Why This Matters", "The pattern repeats every 7 numbers. Since 100 divided by 7 is about 14.28, exactly 14 full cycles occur, each containing one number congruent to 3 mod 7 — confirming our count.", "### Key Takeaway", "Exactly 14 of the first 100 positive integers are congruent to $3 \pmod{7}$. This predictable pattern simplifies counting in modular systems and supports efficient algorithms in programming and mathematical analysis.", "---", "Use this insight to explore more about residue classes, cyclic patterns, and modular arithmetic applications!"]
