Find the smallest positive integer $n$ such that the fourth power of $n$ ends in 0625. - Decision Point

Understanding the Context

In a culture obsessed with precision and evidence, the search for the smallest $ n $ where $ n^4 \equiv 0625 \mod 10000 $ taps into curiosity about number patterns and modulo mathematics. Recent trends show rising interest in computational problems among mobile users who appreciate clean, logical solutions. This is no禳ionable curiosity—users are drawn to elegant answers that align with real-world rules, especially when verified through step-by-step logic. The specificity—ends in 0625—creates a mental anchor, fueling engagement across search, social insights, and educational platforms. As digital literacy grows, so does demand for trustworthy, non-sensational explanations that respect both math and user intelligence.

How to Find the Smallest Positive Integer $ n $ Such That $ n^4 $ Ends in 0625

To solve $ n^4 \mod 10000 = 625 $, we look for integers $ n $ where the last four digits of $ n^4 $ stabilize at 0625. Because powers cycle in modular arithmetic, we examine values of $ n $ modulo 10000, though limited complexity narrows the search. The key lies in testing small integers—from 1 upward—calculating $ n^4 $, and checking the final four digits.

Instead of full large-scale computation, observe that numbers ending in 5 often demonstrate favorable properties with powers: their fourth powers end in 0625 or 0625 in many cases. Let’s test small numbers ending in 5, since 0625 naturally involves a trailing zero and 5-driven cycles.

Key Insights

• $ 5^4 = 625 $ → ends in 0625? No (only 00325)
• $ 15^4 = 50625 $ → ends in 0625 ✅
• Try $ 25^4 = 390625 $ → ends in 0625 ✅
• $ 35^4 = 1500625 $ → ends in 0625 ✅

But $ 15 $ is the smallest. Confirming:
$ 15^2 = 225 $, $ 15^3 = 3375 $, $ 15^4 =