{"status": "success", "data": {"description_md": "Two distinct numbers $a$ and $b$ are chosen randomly from the set $\\{ 2, 2^2, 2^3, \\ldots, 2^{25} \\}$. What is the probability that $\\log_{a} b$ is an integer?\n\n$\\mathrm {(A) } \\frac{2}{25} \\qquad \\mathrm {(B) } \\frac{31}{300} \\qquad \\mathrm {(C) } \\frac{13}{100} \\qquad \\mathrm {(D) } \\frac{7}{50} \\qquad \\mathrm {(E) } \\frac{1}{2}$\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>Two distinct numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  are chosen randomly from the set  <span class=\"katex--inline\">\\{ 2, 2^2, 2^3, \\ldots, 2^{25} \\}</span> . What is the probability that  <span class=\"katex--inline\">\\log_{a} b</span>  is an integer?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm {(A) } \\frac{2}{25} \\qquad \\mathrm {(B) } \\frac{31}{300} \\qquad \\mathrm {(C) } \\frac{13}{100} \\qquad \\mathrm {(D) } \\frac{7}{50} \\qquad \\mathrm {(E) } \\frac{1}{2}</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": "2005 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12A_p24", "prev": "/problem/05_amc12A_p22"}}