Solution: We are looking for integers $ n $ such that $ n \equiv 2 \pmod{5} $ among the first 100 positive integers.
["Finding Integers $ n $ Such That $ n \equiv 2 \pmod{5} $ Among the First 100 Positive Integers", "When studying modular arithmetic, one common problem is identifying integers that satisfy a specific congruence condition—such as finding numbers congruent to $ 2 \mod 5 $ within a fixed range. In this article, we explore how to identify all positive integers $ n \leq 100 $ where $ n \equiv 2 \pmod{5} $, a solution useful in number theory, cyclic patterns, and basic algorithm design.", "---", "### What Does $ n \equiv 2 \pmod{5} $ Mean?", "The expression $ n \equiv 2 \pmod{5} $ means that when $ n $ is divided by 5, the remainder is 2. In other words, $ n $ leaves a specific residue—2—within the modulo 5 system. This pattern repeats every 5 numbers:\n$$\nn = 2, 7, 12, 17, 22, \dots\n$$", "These values form an arithmetic sequence with first term $ a = 2 $ and common difference $ d = 5 $.", "---", "### Step-by-Step: List All Such $ n $ Within the First 100 Integers", "We seek all $ n $ such that:", "$$\nn = 5k + 2 \quad \ ext{and} \quad 1 \leq n \leq 100\n$$", "Substitute $ n = 5k + 2 \leq 100 $ and solve for $ k $:", "$$\n5k + 2 \leq 100 \\n5k \leq 98 \\nk \leq 19.6\n$$", "Since $ k $ must be a non-negative integer, $ k = 0, 1, 2, \dots, 19 $. This gives $ 20 $ distinct values.", "---", "### Explicit List of Solutions", "Substitute $ k = 0 $ through $ k = 19 $ into $ n = 5k + 2 $:", "$$\n\begin{align}\nk = 0 &\Rightarrow n = 2 \\nk = 1 &\Rightarrow n = 7 \\nk = 2 &\Rightarrow n = 12 \\nk = 3 &\Rightarrow n = 17 \\nk = 4 &\Rightarrow n = 22 \\nk = 5 &\Rightarrow n = 27 \\nk = 6 &\Rightarrow n = 32 \\nk = 7 &\Rightarrow n = 37 \\nk = 8 &\Rightarrow n = 42 \\nk = 9 &\Rightarrow n = 47 \\nk = 10 &\Rightarrow n = 52 \\nk = 11 &\Rightarrow n = 57 \\nk = 12 &\Rightarrow n = 62 \\nk = 13 &\Rightarrow n = 67 \\nk = 14 &\Rightarrow n = 72 \\nk = 15 &\Rightarrow n = 77 \\nk = 16 &\Rightarrow n = 82 \\nk = 17 &\Rightarrow n = 87 \\nk = 18 &\Rightarrow n = 92 \\nk = 19 &\Rightarrow n = 97 \\n\end{align}\n$$", "So, the full list contains exactly 20 integers:\n$$\n2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97\n$$", "---", "### Why This Problem Matters", "Identifying numbers in residue classes like $ n \equiv 2 \pmod{5} $ illustrates fundamental concepts in modular arithmetic, which is pivotal in cryptography, error detection, and scheduling algorithms. Understanding how to generate such numbers supports broader mathematical reasoning and computational problem-solving.", "---", "### Final Answer", "There are 20 integers between 1 and 100 that satisfy $ n \equiv 2 \pmod{5} $. They are:", "$$\n2,,7,,12,,17,,22,,27,,32,,37,,42,,47,,52,,57,,62,,67,,72,,77,,82,,87,,92,,97\n$$", "This pattern continues indefinitely, with each term increasing by 5. Recognizing and generating numbers in arithmetic sequences modulo $ m $ is a foundational skill in number theory and applied mathematics.", "---", "Keywords: integers $ n $, $ n \equiv 2 \pmod{5} $, modular arithmetic, first 100 integers, arithmetic sequence, residue classes, number theory, consecutive terms $ n \equiv 2 \pmod{5} $", "Tags: #NumberTheory #ModularArithmetic #MathEducation #IntegerSequences #FractionsAndModuli #Combinatorics"]
