{"status": "success", "data": {"description_md": "How many ordered pairs $(a, b)$ of positive integers satisfy the equation \n$$a\\cdot b + 63 = 20\\cdot \\text{lcm}(a, b) + 12\\cdot\\text{gcd}(a,b),$$\nwhere $\\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\\text{lcm}(a,b)$ denotes their least common multiple?\n\n$\\textbf{(A)} \\text{ 0} \\qquad \\textbf{(B)} \\text{ 2} \\qquad \\textbf{(C)} \\text{ 4} \\qquad \\textbf{(D)} \\text{ 6} \\qquad \\textbf{(E)} \\text{ 8}$", "description_html": "<p>How many ordered pairs  <span class=\"katex--inline\">(a, b)</span>  of positive integers satisfy the equation<br/>\n <span class=\"katex--display\">a\\cdot b + 63 = 20\\cdot \\text{lcm}(a, b) + 12\\cdot\\text{gcd}(a,b),</span> <br/>\nwhere  <span class=\"katex--inline\">\\text{gcd}(a,b)</span>  denotes the greatest common divisor of  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">\\text{lcm}(a,b)</span>  denotes their least common multiple?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A)} \\text{ 0} \\qquad \\textbf{(B)} \\text{ 2} \\qquad \\textbf{(C)} \\text{ 4} \\qquad \\textbf{(D)} \\text{ 6} \\qquad \\textbf{(E)} \\text{ 8}</span> </p>\n<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": 4, "problem_name": "2018 AMC 10B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/18_amc10B_p24", "prev": "/problem/18_amc10B_p22"}}