Solution: We seek the number of positive integers $n \leq 100$ such that $n \equiv 3 \pmod{7}$.
["Finding Positive Integers ≤ 100 Congruent to 3 Modulo 7", "When solving modular arithmetic problems, identifying integers that satisfy specific congruences is a common task. In this article, we explore how to determine the number of positive integers ( n \leq 100 ) such that ( n \equiv 3 \pmod{7} ).", "### Understanding the Congruence", "The condition ( n \equiv 3 \pmod{7} ) means that when ( n ) is divided by 7, the remainder is 3. In other words, ( n ) can be expressed in the form:", "[\nn = 7k + 3\n]", "where ( k ) is a non-negative integer.", "### Finding All Valid ( n \leq 100 )", "We want all such ( n ) satisfying ( n \leq 100 ). Start with the equation:", "[\nn = 7k + 3 \leq 100\n]", "Solving for ( k ):", "[\n7k \leq 97 \quad \Rightarrow \quad k \leq \frac{97}{7} \approx 13.857\n]", "Since ( k ) must be an integer, the largest possible value of ( k ) is 13.", "### Generating the Sequence", "for ( k = 0, 1, 2, \ldots, 13 ):", "- ( k = 0 ): ( n = 3 )\n- ( k = 1 ): ( n = 10 )\n- ( k = 2 ): ( n = 17 )\n- ( k = 3 ): ( n = 24 )\n- ( k = 4 ): ( n = 31 )\n- ( k = 5 ): ( n = 38 )\n- ( k = 6 ): ( n = 45 )\n- ( k = 7 ): ( n = 52 )\n- ( k = 8 ): ( n = 59 )\n- ( k = 9 ): ( n = 66 )\n- ( k = 10 ): ( n = 73 )\n- ( k = 11 ): ( n = 80 )\n- ( k = 12 ): ( n = 87 )\n- ( k = 13 ): ( n = 94 )", "Note that when ( k = 14 ), ( n = 7(14) + 3 = 101 ), which exceeds 100, so it is excluded.", "### Counting the Valid Values", "We counted integer values of ( k ) from 0 to 13, inclusive—this is a total of:", "[\n13 - 0 + 1 = 14\n]", "Therefore, there are 14 positive integers less than or equal to 100 that satisfy ( n \equiv 3 \pmod{7} ).", "### Summary", "- Condition: ( n \equiv 3 \pmod{7} )\n- General form: ( n = 7k + 3 )\n- Constraint: ( n \leq 100 )\n- Maximum ( k ): 13\n- Count: ( k = 0 ) to ( 13 ) → 14 values", "This approach—expressing the solution via modular arithmetic and systematically counting valid integers—is a clean and reliable method widely applicable in number theory and algorithm design.", "---", "Keywords:\npositive integers ( n \leq 100 ), ( n \equiv 3 \pmod{7} ), modular arithmetic solution, counting integers, arithmetic progression, number theory, modular congruence count."]
