Thus, \( k = 5m + 2 \), so \( n = 10(5m + 2) + 4 = 50m + 24 \). The smallest positive \( n \) is \( n = 24 \). Check: \( 24^2 = 576 \), ends in 76 — valid.

["Understanding Linear Expressions and Perfect Squares: How ( k = 5m + 2 ) Yields Perfect Squares Ending in 76", "Mathematics often reveals elegant patterns through simple algebraic identities. One such pattern involves a straightforward linear relationship: given ( k = 5m + 2 ), this expression naturally generates perfect squares ending in 76 — specifically, ( n = 10k + 4 = 50m + 24 ). In this article, we explore how this transformation works, why these squares end in 76, and verify the smallest solution with a simple calculation.", "---", "### The Formula at a Glance", "Start with the linear expression:\n[\nk = 5m + 2\n]\nUsing this, substitute into the square expression:\n[\nn = 10k + 4 = 10(5m + 2) + 4 = 50m + 20 + 4 = 50m + 24\n]\nSo, for any integer ( m ), ( n = 50m + 24 ) produces a perfect square ending in 76.", "---", "### Why Do Squares End in 76?", "Perfect squares ending in specific digits depend on their last two digits. For ( n = 50m + 24 ), note that:", "- ( 50m ) ends in 00, so\n- Adding 24 shifts the ending to 24, regardless of ( m ), as long as ( m ) is an integer.", "Thus, all numbers of the form ( 50m + 24 ) produce squares ending in 76 — for example, ( 24^2 = 576 ), ( 74^2 = 5476 ), and ( 124^2 = 15376 ), each ending in 76.", "---", "### Finding the Smallest Positive ( n )", "To find the smallest positive ( n ) of this form, simply plug in ( m = 0 ):\n[\nn = 50(0) + 24 = 24\n]", "Now verify if ( 24 ) is indeed a perfect square ending in 76:\n[\n24^2 = 576 \quad \ ext{(ends in 76)} \quad \ ext{✓}\n]\nNo smaller positive ( n ) satisfies both conditions, so the smallest solution is:\n[\nn = 24\n]", "---", "### Final Check: Square of 24", "Let’s double-check:\n[\n24 \ imes 24 = 576\n]\nIndeed, ( 576 ) ends with 76, confirming the correctness.", "---", "### Conclusion", "The relationship ( k = 5m + 2 ) generates perfect squares through the linear formula ( n = 50m + 24 ). This clever substitution reveals a systematic source of perfect squares ending in 76, a property both mathematically intriguing and computationally easy to verify. The smallest such square comes from ( m = 0 ), yielding ( n = 24 ), whose square, ( 576 ), confirms the pattern’s validity.", "This example beautifully demonstrates how algebra and digit analysis converge to uncover structured number patterns — ideal for learners exploring number theory and patterns in mathematics.", "---", "Keywords:\n( n = 50m + 24 ), perfect squares ending in 76, ( k = 5m + 2 ), smallest square ending in 76, algebra and digit patterns, mathematical verification, ( 24^2 = 576 )"]