This implies $x \equiv \pm5 \pmod{10}$. Testing $x = 5$: $5^2 = 25$ ends in 25.

["Understanding the Modular Condition: $x \equiv \pm5 \pmod{10}$ and the Test with $x = 5$", "When studying number theory and modular arithmetic, one frequently encounters conditions expressed in congruences like $x \equiv \pm5 \pmod{10}$. This convention reveals important patterns about numbers and their squares — especially how their last digits behave.", "### What Does $x \equiv \pm5 \pmod{10}$ Mean?", "The expression $x \equiv \pm5 \pmod{10}$ mathematically means that when $x$ is divided by 10, the remainder is either $5$ or $-5$. Since modular arithmetic simplifies negatives (e.g., $-5 \equiv 5 \pmod{10}$), this condition is equivalent to:", "$$\nx \equiv 5 \pmod{10} \quad \ ext{or} \quad x \equiv 5 \pmod{10}\n$$", "This implies that $x$ ends in either 5.", "### Why Numbers Ending in 5 Are Special Squares", "Consider squaring any number ending in 5 — such as 5, 15, 25, etc. The result always ends in 25. For example:", "- $5^2 = 25$\n- $15^2 = 225$\n- $25^2 = 625$\n- $35^2 = 1225$", "Notice the pattern: the final two digits of the square are 25, meaning:", "$$\nx^2 \equiv 25 \pmod{100}\n$$", "In particular, when $x \equiv 5 \pmod{10}$, $x^2 \equiv 25 \pmod{100}$, confirming the ending digits.", "### Testing $x = 5$", "Let $x = 5$:", "- $5 \equiv 5 \pmod{10}$, so $x$ satisfies the condition.\n- Compute $5^2 = 25$, which indeed ends in 25.", "This simple test confirms that numbers congruent to $5$ or $5 \pmod{10}$ (i.e., those ending in 5) produce squares ending in 25 — a consistent rule supported by modular arithmetic.", "### Broader Applications in Problem Solving", "Recognizing this pattern is useful in:", "- Identifying candidates for squares ending in certain digits\n- Solving congruence systems in number theory\n- Verifying properties of integers in integer programming or coding theory", "---", "Key Takeaways:", "- $x \equiv \pm5 \pmod{10}$ means $x$ ends in 5.\n- Squaring such numbers yields squares ending in 25.\n- Testing $x = 5$ confirms $5^2 = 25$, validating the condition.\n- This modular insight simplifies analysis of number patterns and digital endings.", "Understanding such modular relationships deepens insight into number behavior — essential for learners, educators, and enthusiasts exploring number theory.", "---", "Explore more about modular arithmetic, digit patterns, and integer properties to unlock powerful problem-solving tools in mathematics."]