{"status": "success", "data": {"description_md": "Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3, ... , 2010\\}$. What is the probability that $abc + ab + a$ is divisible by $3$?\n\n$\\textbf{(A)}\\ \\dfrac{1}{3} \\qquad \\textbf{(B)}\\ \\dfrac{29}{81} \\qquad \\textbf{(C)}\\ \\dfrac{31}{81} \\qquad \\textbf{(D)}\\ \\dfrac{11}{27} \\qquad \\textbf{(E)}\\ \\dfrac{13}{27}$", "description_html": "<p>Positive integers <span class=\"katex--inline\">a</span>, <span class=\"katex--inline\">b</span>, and <span class=\"katex--inline\">c</span> are randomly and independently selected with replacement from the set <span class=\"katex--inline\">\\{1, 2, 3, ... , 2010\\}</span>. What is the probability that <span class=\"katex--inline\">abc + ab + a</span> is divisible by <span class=\"katex--inline\">3</span>?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}\\ \\dfrac{1}{3} \\qquad \\textbf{(B)}\\ \\dfrac{29}{81} \\qquad \\textbf{(C)}\\ \\dfrac{31}{81} \\qquad \\textbf{(D)}\\ \\dfrac{11}{27} \\qquad \\textbf{(E)}\\ \\dfrac{13}{27}</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2010 AMC 10B Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/10_amc10B_p19", "prev": "/problem/10_amc10B_p17"}}