{"status": "success", "data": {"description_md": "Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3,\\ldots, 2010\\}$. What is the probability that $abc + ab + a$ is divisible by $3$?\n\n$\\textbf{(A)}\\ \\dfrac{1}{3} \\qquad \\textbf{(B)}\\ \\dfrac{29}{81} \\qquad \\textbf{(C)}\\ \\dfrac{31}{81} \\qquad \\textbf{(D)}\\ \\dfrac{11}{27} \\qquad \\textbf{(E)}\\ \\dfrac{13}{27}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "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,\\ldots, 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;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2010 AMC 12B Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/10_amc12B_p17", "prev": "/problem/10_amc12B_p15"}}