{"status": "success", "data": {"description_md": "The number of ordered $5$-tuples $(a,b,c,d,e)$ of positive integers where $a$, $b$, $c$, and $d$ are perfect squares and $e$ is a perfect cube, and $abcde= 645^{245}$ can be expressed in the form $$ p_1^{q_1} \\cdot p_2^{q_2} \\cdot p_3^{q_3} \\cdots p_n^{q_n}$$ where the $p_i$ are distinct prime numbers and the $q_i$ are positive integer exponents.\n\nCompute $$ (p_1 + p_2 + \\cdots + p_n) + (q_1 + q_2 + \\cdots + q_n). $$", "description_html": "<p>The number of ordered <span class=\"katex--inline\">5</span>-tuples <span class=\"katex--inline\">(a,b,c,d,e)</span> of positive integers where <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 perfect squares and <span class=\"katex--inline\">e</span> is a perfect cube, and <span class=\"katex--inline\">abcde= 645^{245}</span> can be expressed in the form <span class=\"katex--display\"> p_1^{q_1} \\cdot p_2^{q_2} \\cdot p_3^{q_3} \\cdots p_n^{q_n}</span> where the <span class=\"katex--inline\">p_i</span> are distinct prime numbers and the <span class=\"katex--inline\">q_i</span> are positive integer exponents.</p>&#10;<p>Compute <span class=\"katex--display\"> (p_1 + p_2 + \\cdots + p_n) + (q_1 + q_2 + \\cdots + q_n). </span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Mock Euclid 2025 - Problem 8 Part B", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}