{"problem":{"name":"H. Riana and Humongous Numbers","description":{"content":"Riana got really bored one day and decided to play with some integers. She started with the positive integer $N$. She then noticed that $N$ was too small and boring. Suddenly, she got the idea to crea","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":1000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF10255H"},"statements":[{"statement_type":"Markdown","content":"Riana got really bored one day and decided to play with some integers. She started with the positive integer $N$. She then noticed that $N$ was too small and boring. Suddenly, she got the idea to create a really big integer. To do so, she multiplied all of the positive divisors of $N$ and got the integer $M$. After getting the value of $M$, she challenged her friend Cisco to determine the value of the original integer $N$ given $M$. Help Cisco find the value of $N$.\n\nThe input contains one line containing the integer $M$. It is guaranteed that $M$ is a positive integer less than $10^(15)$.\n\nOutput the original integer $N$ if possible. Otherwise, output $-1$.\n\nIn the first test case, the positive divisors of 4 are 1, 2, and 4. The product is 8. \n\nIn the second test case, there is no integer whose product of its positive divisors is 4.\n\n## Input\n\nThe input contains one line containing the integer $M$. It is guaranteed that $M$ is a positive integer less than $10^(15)$.\n\n## Output\n\nOutput the original integer $N$ if possible. Otherwise, output $-1$.\n\n[samples]\n\n## Note\n\nIn the first test case, the positive divisors of 4 are 1, 2, and 4. The product is 8. In the second test case, there is no integer whose product of its positive divisors is 4.","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"**Definitions**  \nLet $ M \\in \\mathbb{Z}^+ $ be the given product of all positive divisors of some positive integer $ N $.  \n\n**Constraints**  \n$ 1 \\leq M < 10^{15} $\n\n**Objective**  \nFind $ N \\in \\mathbb{Z}^+ $ such that the product of all positive divisors of $ N $ equals $ M $, i.e.,  \n$$\n\\prod_{d \\mid N} d = M\n$$  \nIf no such $ N $ exists, output $ -1 $.  \n\n**Key Identity**  \nIf $ N $ has $ d(N) $ positive divisors, then:  \n$$\n\\prod_{d \\mid N} d = N^{d(N)/2}\n$$  \nThus, we require:  \n$$\nN^{d(N)/2} = M\n$$","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF10255H","tags":[],"sample_group":[],"created_at":"2026-03-03 11:00:39"}}