Between 24 and 26, the smaller is 24. Therefore, the smallest positive integer whose square ends in 76 is \( \boxed{24} \).

["Understanding the Smallest Positive Integer Whose Square Ends in 76", "When solving math puzzles like finding the smallest positive integer whose square ends in 76, many wonder: between 24 and 26, the smaller being 24—does that prove 24 is the answer? Depending on careful analysis, yes, and here’s why.", "### Why 24 is the Smallest Positive Integer with a Square Ending in 76\nWe begin by computing the square of 24:\n[\n24^2 = 576\n]\nIndeed, 576 ends in 76. Next, we check if any smaller positive integer—specifically between 1 and 23—also satisfies this condition.", "We know the problem states that between 24 and 26, the smaller value is 24—suggesting 24 is the first in this range. More importantly, we analyze integers ending or nearly ending in digits that, when squared, may produce a number ending in 76.", "Squares ending in 76 impose a strong constraint:\n[\nn^2 \equiv 76 \pmod{100}\n]\nSo, we search for integers ( n ) such that:\n[\nn^2 \mod 100 = 76\n]", "Rather than test every number from 1 to 100, we can refine our search by noting:", "- The last digit of ( n ) must produce a square ending in 6.\n Only numbers ending in 4 or 6 have squares ending in 6:\n ( 4^2 = 16 ), ( 6^2 = 36 )", "- Testing numbers ending in 4 or 6 between 1 and 25:\n ( 14^2 = 196 )\n ( 16^2 = 256 )\n ( 24^2 = 576 ) ← ends in 76 ✅\n No smaller number between 1 and 13 produces a square ending in 76.", "Further, modulo analysis confirms:\nWe seek ( n^2 \equiv 76 \pmod{100} ).\nChecking modulo 4 and modulo 25:\n- ( n^2 \equiv 0 \pmod{4} ) (since 76 is divisible by 4) → ( n ) even\n- ( n^2 \equiv 76 \equiv 1 \pmod{25} ) → Solve ( n^2 \equiv 1 \pmod{25} ) → solutions ( n \equiv \pm1 \pmod{25} )", "Combining these with Chinese Remainder Theorem and testing small solutions confirms the smallest such positive integer is ( n = 24 ).", "### Final Conclusion\nBetween 24 and 26, 24 is the smallest integer satisfying ( n^2 ) ending in 76, confirming:\n[\n\boxed{24}\n]\nThis small but precise result highlights how number patterns and modular arithmetic uncover elegant solutions in mathematics. Whether for puzzles or deeper exploration, understanding such properties enriches problem-solving skills.", "---", "Keywords: smallest positive integer square ending in 76, 24 vs 26, 24 squared 576, math puzzle solution, last two digits square, modular arithmetic in squares, math problem explanation.\nMeta Description: Discover why 24 is the smallest positive integer whose square ends in 76 through step-by-step reasoning, modular analysis, and verification of all smaller candidates."]