{"status": "success", "data": {"description_md": "Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.\n\n$$abcd=2^6\\cdot 3^9\\cdot 5^7$$\n\n$$\\text{lcm}(a,b)=2^3\\cdot 3^2\\cdot 5^3$$\n\n$$\\text{lcm}(a,c)=2^3\\cdot 3^3\\cdot 5^3$$\n\n$$\\text{lcm}(a,d)=2^3\\cdot 3^3\\cdot 5^3$$\n\n$$\\text{lcm}(b,c)=2^1\\cdot 3^3\\cdot 5^2$$\n\n$$\\text{lcm}(b,d)=2^2\\cdot 3^3\\cdot 5^2$$\n\n$$\\text{lcm}(c,d)=2^2\\cdot 3^3\\cdot 5^2$$<br>What is $\\text{gcd}(a,b,c,d)$?\n\n$\\textbf{(A)}~30\\qquad\\textbf{(B)}~45\\qquad\\textbf{(C)}~3\\qquad\\textbf{(D)}~15\\qquad\\textbf{(E)}~6$\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>Suppose that  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> ,  <span class=\"katex--inline\">c</span>  and  <span class=\"katex--inline\">d</span>  are positive integers satisfying all of the following relations.</p>&#10;<p> <span class=\"katex--display\">abcd=2^6\\cdot 3^9\\cdot 5^7</span> </p>&#10;<p> <span class=\"katex--display\">\\text{lcm}(a,b)=2^3\\cdot 3^2\\cdot 5^3</span> </p>&#10;<p> <span class=\"katex--display\">\\text{lcm}(a,c)=2^3\\cdot 3^3\\cdot 5^3</span> </p>&#10;<p> <span class=\"katex--display\">\\text{lcm}(a,d)=2^3\\cdot 3^3\\cdot 5^3</span> </p>&#10;<p> <span class=\"katex--display\">\\text{lcm}(b,c)=2^1\\cdot 3^3\\cdot 5^2</span> </p>&#10;<p> <span class=\"katex--display\">\\text{lcm}(b,d)=2^2\\cdot 3^3\\cdot 5^2</span> </p>&#10;<p> <span class=\"katex--display\">\\text{lcm}(c,d)=2^2\\cdot 3^3\\cdot 5^2</span> <br/>What is  <span class=\"katex--inline\">\\text{gcd}(a,b,c,d)</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}~30\\qquad\\textbf{(B)}~45\\qquad\\textbf{(C)}~3\\qquad\\textbf{(D)}~15\\qquad\\textbf{(E)}~6</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": 5, "problem_name": "2023 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12B_p25", "prev": "/problem/23_amc12B_p23"}}