E. Mod Mod Mod

Codeforces

IDCF889E

Time2000ms

Memory256MB

Difficulty—

binary searchdpmath

English · Original

Chinese · Translation

Formal · Original

You are given a sequence of integers _a_1, _a_2, ..., _a__n_. Let , and for 1 ≤ _i_ < _n_. Here, denotes the modulus operation. Find the maximum value of _f_(_x_, 1) over all nonnegative integers _x_. ## Input The first line contains a single integer _n_ (1 ≤ _n_ ≤ 200000) — the length of the sequence. The second lines contains _n_ integers _a_1, _a_2, ..., _a__n_ (1 ≤ _a__i_ ≤ 1013) — the elements of the sequence. ## Output Output a single integer — the maximum value of _f_(_x_, 1) over all nonnegative integers _x_. [samples] ## Note In the first example you can choose, for example, _x_ = 19. In the second example you can choose, for example, _x_ = 3 or _x_ = 2.

给定一个整数序列 #cf_span[a1, a2, ..., an]。定义函数 f(x, i) 如下：f(x, n) = x，且对于 #cf_span[1 ≤ i < n]，有 f(x, i) = f(x, i+1) \% a_i。这里，\% 表示取模运算。求在所有非负整数 #cf_span[x] 中，#cf_span[f(x, 1)] 的最大值。第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 200000]) —— 序列的长度。第二行包含 #cf_span[n] 个整数 #cf_span[a1, a2, ..., an] (#cf_span[1 ≤ ai ≤ 1013]) —— 序列的元素。请输出一个整数 —— #cf_span[f(x, 1)] 在所有非负整数 #cf_span[x] 中的最大值。在第一个例子中，你可以选择，例如，#cf_span[x = 19]。在第二个例子中，你可以选择，例如，#cf_span[x = 3] 或 #cf_span[x = 2]。 ## Input 第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 200000]) —— 序列的长度。第二行包含 #cf_span[n] 个整数 #cf_span[a1, a2, ..., an] (#cf_span[1 ≤ ai ≤ 1013]) —— 序列的元素。 ## Output 请输出一个整数 —— #cf_span[f(x, 1)] 在所有非负整数 #cf_span[x] 中的最大值。 [samples] ## Note 在第一个例子中，你可以选择，例如，#cf_span[x = 19]。在第二个例子中，你可以选择，例如，#cf_span[x = 3] 或 #cf_span[x = 2]。

Let $ n \in \mathbb{N} $, $ n \geq 1 $, and let $ a_1, a_2, \dots, a_n \in \mathbb{N} $ with $ a_i \geq 1 $. Define the function $ f : \mathbb{Z}_{\geq 0} \times \{1, 2, \dots, n\} \to \mathbb{Z}_{\geq 0} $ recursively by: $$ f(x, i) = \begin{cases} x \bmod a_i & \text{if } i = n, \\ f(x \bmod a_i, i+1) & \text{if } 1 \leq i < n. \end{cases} $$ Find: $$ \max_{x \in \mathbb{Z}_{\geq 0}} f(x, 1) $$

Samples

Input #1

2
10 5

Output #1

Input #2

5
5 4 3 2 1

Output #2

Input #3

4
5 10 5 10

Output #3

API Response (JSON)

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  "problem": {
    "name": "E. Mod Mod Mod",
    "description": {
      "content": "You are given a sequence of integers _a_1, _a_2, ..., _a__n_. Let , and for 1 ≤ _i_ < _n_. Here, denotes the modulus operation. Find the maximum value of _f_(_x_, 1) over all nonnegative integers _x_.",
      "description_type": "Markdown"
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    "platform": "Codeforces",
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      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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    "is_sync": true,
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      "content": "You are given a sequence of integers _a_1, _a_2, ..., _a__n_. Let , and for 1 ≤ _i_ < _n_. Here, denotes the modulus operation. Find the maximum value of _f_(_x_, 1) over all nonnegative integers _x_....",
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      "statement_type": "Markdown",
      "content": "给定一个整数序列 #cf_span[a1, a2, ..., an]。定义函数 f(x, i) 如下：f(x, n) = x，且对于 #cf_span[1 ≤ i < n]，有 f(x, i) = f(x, i+1)  \\%  a_i。这里，\\% 表示取模运算。求在所有非负整数 #cf_span[x] 中，#cf_span[f(x, 1)] 的最大值。\n\n第一行包含一个整数 #cf_span[n]...",
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      "content": "Let $ n \\in \\mathbb{N} $, $ n \\geq 1 $, and let $ a_1, a_2, \\dots, a_n \\in \\mathbb{N} $ with $ a_i \\geq 1 $.\n\nDefine the function $ f : \\mathbb{Z}_{\\geq 0} \\times \\{1, 2, \\dots, n\\} \\to \\mathbb{Z}_{\\g...",
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