What is the ones digit of the sum
$$\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \dots + \lfloor \sqrt{2025} \rfloor?$$(Recall that $\lfloor x \rfloor$ represents the greatest integer less than or equal to $x$.)
$\textbf{(A) } 1\qquad\textbf{(B) } 2 \qquad\textbf{(C) }3\qquad\textbf{(D) } 5 \qquad\textbf{(E) } 8$
Difficulty: 27/100 - Not too hard once a pattern is recognized.
Core Concepts: pattern recognition, cycles, computation
Challenges: It may feel tempting to brute force especially if the pattern isn't seen.
Techniques: Finding a pattern in how many times a specific integer appears and executing a computation.
Error-prone Steps: Forgetting to add the $\sqrt{2025}$ at the end of the sum.
Ideal Time:
Experienced: ≤ 1-2 min
Intermediate: ≤ 3 min
Beginner: ≤ 8 min