{"problem":{"name":"E. Counting Arrays","description":{"content":"You are given two positive integer numbers _x_ and _y_. An array _F_ is called an __y_\\-factorization_ of _x_ iff the following conditions are met: *   There are _y_ elements in _F_, and all of them ","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":3000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF893E"},"statements":[{"statement_type":"Markdown","content":"You are given two positive integer numbers _x_ and _y_. An array _F_ is called an __y_\\-factorization_ of _x_ iff the following conditions are met:\n\n*   There are _y_ elements in _F_, and all of them are integer numbers;\n*   .\n\nYou have to count the number of pairwise distinct arrays that are _y_\\-factorizations of _x_. Two arrays _A_ and _B_ are considered different iff there exists at least one index _i_ (1 ≤ _i_ ≤ _y_) such that _A__i_ ≠ _B__i_. Since the answer can be very large, print it modulo 109 + 7.\n\n## Input\n\nThe first line contains one integer _q_ (1 ≤ _q_ ≤ 105) — the number of testcases to solve.\n\nThen _q_ lines follow, each containing two integers _x__i_ and _y__i_ (1 ≤ _x__i_, _y__i_ ≤ 106). Each of these lines represents a testcase.\n\n## Output\n\nPrint _q_ integers. _i_\\-th integer has to be equal to the number of _y__i_\\-factorizations of _x__i_ modulo 109 + 7.\n\n[samples]\n\n## Note\n\nIn the second testcase of the example there are six _y_\\-factorizations:\n\n*   { - 4,  - 1};\n*   { - 2,  - 2};\n*   { - 1,  - 4};\n*   {1, 4};\n*   {2, 2};\n*   {4, 1}.","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"给你两个正整数 $x$ 和 $y$。一个数组 $F$ 被称为 $x$ 的一个 _$y$-因子分解_，当且仅当满足以下条件：\n\n你需要计算有多少个互不相同的数组是 $x$ 的 $y$-因子分解。两个数组 $A$ 和 $B$ 被认为不同，当且仅当存在至少一个下标 $i$（$1 ≤ i ≤ y$）使得 $A_i ≠ B_i$。由于答案可能非常大，请对 $10^9 + 7$ 取模输出。\n\n第一行包含一个整数 $q$（$1 ≤ q ≤ 10^5$）——表示待解决的测试用例数量。\n\n接下来 $q$ 行，每行包含两个整数 $x_i$ 和 $y_i$（$1 ≤ x_i, y_i ≤ 10^6$）。每行代表一个测试用例。\n\n请输出 $q$ 个整数。第 $i$ 个整数应等于 $x_i$ 的 $y_i$-因子分解的数量，对 $10^9 + 7$ 取模。\n\n在示例的第二个测试用例中，有六个 $y$-因子分解：\n\n## Input\n\n第一行包含一个整数 $q$（$1 ≤ q ≤ 10^5$）——表示待解决的测试用例数量。接下来 $q$ 行，每行包含两个整数 $x_i$ 和 $y_i$（$1 ≤ x_i, y_i ≤ 10^6$）。每行代表一个测试用例。\n\n## Output\n\n请输出 $q$ 个整数。第 $i$ 个整数应等于 $x_i$ 的 $y_i$-因子分解的数量，对 $10^9 + 7$ 取模。\n\n[samples]\n\n## Note\n\n在示例的第二个测试用例中，有六个 $y$-因子分解：$\\{-4, -1\\}$；$\\{-2, -2\\}$；$\\{-1, -4\\}$；$\\{1, 4\\}$；$\\{2, 2\\}$；$\\{4, 1\\}$。","is_translate":true,"language":"Chinese"},{"statement_type":"Markdown","content":"**Definitions**  \nLet $ q \\in \\mathbb{Z} $ be the number of test cases.  \nFor each test case $ i \\in \\{1, \\dots, q\\} $, let $ x_i, y_i \\in \\mathbb{Z}^+ $ be given integers.  \n\nA **$ y_i $-factorization** of $ x_i $ is a $ y_i $-tuple $ (a_1, a_2, \\dots, a_{y_i}) \\in (\\mathbb{Z}^+)^{y_i} $ such that:  \n$$\n\\prod_{j=1}^{y_i} a_j = x_i\n$$\n\nTwo $ y_i $-factorizations are **distinct** if they differ in at least one component.\n\n**Constraints**  \n1. $ 1 \\le q \\le 10^5 $  \n2. For each test case $ i $: $ 1 \\le x_i, y_i \\le 10^6 $\n\n**Objective**  \nFor each test case $ i $, compute the number of distinct $ y_i $-factorizations of $ x_i $, modulo $ 10^9 + 7 $:  \n$$\n\\left| \\left\\{ (a_1, \\dots, a_{y_i}) \\in (\\mathbb{Z}^+)^{y_i} \\ \\middle|\\ \\prod_{j=1}^{y_i} a_j = x_i \\right\\} \\right| \\mod (10^9 + 7)\n$$","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF893E","tags":["combinatorics","dp","math","number theory"],"sample_group":[["2\n6 3\n4 2","36\n6"]],"created_at":"2026-03-03 11:00:39"}}