Solution:** We want integers \( n \) such that \( 1 \leq n \leq 150 \) and \( n \equiv 3 \pmod{7} \).

["### Find All Integers ( n ) Between 1 and 150 with ( n \equiv 3 \pmod{7} )", "If you're looking for all integers ( n ) in the range ( 1 \leq n \leq 150 ) that satisfy the modular condition ( n \equiv 3 \pmod{7} ), you’ve come to the right place.", "#### What Does ( n \equiv 3 \pmod{7} ) Mean?", "This congruence means that when ( n ) is divided by 7, the remainder is 3. In other words, ( n ) can be written in the form:", "[\nn = 7k + 3\n]", "where ( k ) is an integer.", "#### How to Find All Valid ( n ) in the Range ( 1 \leq n \leq 150 )", "To find all such integers, substitute values of ( k ) into ( n = 7k + 3 ) and determine which values keep ( n ) within the desired range.", "Start solving for ( k ):", "[\n1 \leq 7k + 3 \leq 150\n]", "Subtract 3 from all parts:", "[\n-2 \leq 7k \leq 147\n]", "Then divide by 7:", "[\n-\frac{2}{7} \leq k \leq 21\n]", "Since ( k ) must be an integer, the smallest possible ( k ) is 0, and the largest is 21.", "#### Generate All Valid ( n )", "For ( k = 0, 1, 2, \dots, 21 ), compute:", "[\nn = 7k + 3\n]", "This gives the sequence:", "- ( 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 )\n- ( k = 14 ): ( n = 101 )\n- ( k = 15 ): ( n = 108 )\n- ( k = 16 ): ( n = 115 )\n- ( k = 17 ): ( n = 122 )\n- ( k = 18 ): ( n = 129 )\n- ( k = 19 ): ( n = 136 )\n- ( k = 20 ): ( n = 143 )\n- ( k = 21 ): ( n = 150 )", "#### Summary of All Valid ( n )", "There are 22 integers between 1 and 150 such that ( n \equiv 3 \pmod{7} ), starting from 3 and increasing by 7 each time:", "[\n3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150\n]", "#### Why This Matters", "Finding integers satisfying modular congruences is essential in number theory, computer science, cryptography, and algorithm design. This simple yet powerful pattern helps solve problems related to periodicity, hash functions, and modular arithmetic applications.", "#### How to Use This List", "- Filter arrays or datasets for values satisfying ( n \equiv 3 \pmod{7} )\n- Identify conflict-free schedules or checkpoints spaced every 7 units starting at 3\n- Solve modular constraints in optimization or constraint programming", "---", "Keywords: integers ( n ) such that ( 1 \leq n \leq 150 ), ( n \equiv 3 \pmod{7} ), modular arithmetic, congruence solutions, range 1 to 150, list of n ≡ 3 mod 7", "If you want to verify or compute values efficiently, simply iterate over ( k = 0 ) to ( 21 ), compute ( n = 7k + 3 ), and collect those within bounds. This fundamental pattern ensures you never miss valid values.", "---", "Related Topics:\n- How to find numbers ( n \equiv a \pmod{b} ) in a range\n- Modular arithmetic examples\n- Practical uses of congruences in computer science\n- Counting integers in arithmetic sequences", "---", "Final Note:\nAll integers ( n ) between 1 and 150 satisfying ( n \equiv 3 \pmod{7} ) form the 22-element arithmetic sequence starting at 3 with common difference 7.\nUse this list to satisfy modular constraints confidently."]