{"status": "success", "data": {"description_md": "Bob has a machine that takes two numbers $a$, $b$ on a whiteboard and replaces them with $\\gcd(a^2+1,b^2+1)$ and $\\text{lcm}(a,b)$ (order doesn't matter). Find the number of unordered pairs $(a,b)$ with $1\\le a,b\\le 1234$ such that Bob's machine replaces $(a,b)$ with $(a,b)$.", "description_html": "<p>Bob has a machine that takes two numbers <span class=\"katex--inline\">a</span>, <span class=\"katex--inline\">b</span> on a whiteboard and replaces them with <span class=\"katex--inline\">\\gcd(a^2+1,b^2+1)</span> and <span class=\"katex--inline\">\\text{lcm}(a,b)</span> (order doesn&#8217;t matter). Find the number of unordered pairs <span class=\"katex--inline\">(a,b)</span> with <span class=\"katex--inline\">1\\le a,b\\le 1234</span> such that Bob&#8217;s machine replaces <span class=\"katex--inline\">(a,b)</span> with <span class=\"katex--inline\">(a,b)</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Grinding Aces Math Exam 2025 - Problem 3", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}