#### 420Question: How many integers between 1 and 500 are congruent to 2 mod 7?
["How Many Integers Between 1 and 500 Are Congruent to 2 Mod 7?", "When exploring patterns in numbers, one common and insightful question is: How many integers between 1 and 500 are congruent to 2 modulo 7? This type of query helps unlock modular arithmetic and number distribution fundamentals.", "### Understanding “Congruent to 2 Mod 7”", "In mathematics, saying an integer n is congruent to 2 modulo 7 means:", "[\nn \equiv 2 \pmod{7}\n]", "This means n leaves a remainder of 2 when divided by 7. In other words, n can be expressed in the form:", "[\nn = 7k + 2\n]", "for some integer k.", "### Find All Integers Between 1 and 500 Satisfying the Condition", "We are looking for all integers n such that:", "[\n1 \leq n \leq 500 \quad \ ext{and} \quad n = 7k + 2\n]", "We substitute and solve for k:", "[\n1 \leq 7k + 2 \leq 500\n]", "Subtract 2 from all parts:", "[\n-1 \leq 7k \leq 498\n]", "Divide through by 7:", "[\n-\frac{1}{7} \leq k \leq 71.142...\n]", "Since k must be an integer, valid values for k range from 0 to 71.", "### Count the Valid Values of k", "The integer k ranges from 0 to 71 inclusive. The total number of integers in this range is:", "[\n71 - 0 + 1 = 72\n]", "### Verify Boundary Values", "- When k = 0: ( n = 7(0) + 2 = 2 ) (valid, between 1 and 500)\n- When k = 71: ( n = 7(71) + 2 = 497 + 2 = 499 ) (still within range)", "Thus, all 72 values are within the specified interval.", "### Conclusion", "There are exactly 72 integers between 1 and 500 that are congruent to 2 modulo 7. This result showcases how modular patterns evenly distribute over intervals, making this a useful arithmetic insight.", "---", "Key Takeaways:", "- Use the form ( n = 7k + 2 ) for numbers ≡ 2 mod 7\n- Solve the inequality ( 1 \leq 7k + 2 \leq 500 ) to find permissible k\n- Count integer values of k from 0 to 71 for 72 solutions\n- This method applies broadly to problems involving modular arithmetic", "---", "See also:\n- Modular arithmetic explained\n- Counting numbers in arithmetic progressions\n- Applications of congruences in number theory", "---", "Keywords: \ncongruentto2mod7, #integersbetween1and500, #mod7#, #countingnumbersmodulo7, #numberpatterns, #modulararithmetic\nMeta description:\nDiscover how many integers between 1 and 500 satisfy ( n \equiv 2 \pmod{7} ) using modular arithmetic. Find exact count and step-by-step explanation."]
