{"status": "success", "data": {"description_md": "For every integer $n\\ge2$, let $\\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\\text{pow}(144)=\\text{pow}(2^4\\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides\n\n$$$\\prod_{n=2}^{5300}\\text{pow}(n)$?$$\n\n$\\textbf{(A)}\\ 74 \\qquad \\textbf{(B)}\\ 75 \\qquad \\textbf{(C)}\\ 76 \\qquad \\textbf{(D)}\\ 77 \\qquad \\textbf{(E)}\\ 78$\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>For every integer  <span class=\"katex--inline\">n\\ge2</span> , let  <span class=\"katex--inline\">\\text{pow}(n)</span>  be the largest power of the largest prime that divides  <span class=\"katex--inline\">n</span> . For example  <span class=\"katex--inline\">\\text{pow}(144)=\\text{pow}(2^4\\cdot3^2)=3^2</span> . What is the largest integer  <span class=\"katex--inline\">m</span>  such that  <span class=\"katex--inline\">2010^m</span>  divides</p>&#10;<p>$ <span class=\"katex--display\">\\prod_{n=2}^{5300}\\text{pow}(n)$?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 74 \\qquad \\textbf{(B)}\\ 75 \\qquad \\textbf{(C)}\\ 76 \\qquad \\textbf{(D)}\\ 77 \\qquad \\textbf{(E)}\\ 78</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": "2010 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/10_amc12B_p24"}}