English · Original
Chinese · Translation
Formal · Original
You have an array _a_ with length _n_, you can perform operations. Each operation is like this: choose two **adjacent** elements from _a_, say _x_ and _y_, and replace one of them with _gcd_(_x_, _y_), where _gcd_ denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).
What is the minimum number of operations you need to make all of the elements equal to 1?
## Input
The first line of the input contains one integer _n_ (1 ≤ _n_ ≤ 2000) — the number of elements in the array.
The second line contains _n_ space separated integers _a_1, _a_2, ..., _a__n_ (1 ≤ _a__i_ ≤ 109) — the elements of the array.
## Output
Print _\-1_, if it is impossible to turn all numbers to 1. Otherwise, print the minimum number of operations needed to make all numbers equal to 1.
[samples]
## Note
In the first sample you can turn all numbers to 1 using the following 5 moves:
* \[2, 2, 3, 4, 6\].
* \[2, 1, 3, 4, 6\]
* \[2, 1, 3, 1, 6\]
* \[2, 1, 1, 1, 6\]
* \[1, 1, 1, 1, 6\]
* \[1, 1, 1, 1, 1\]
We can prove that in this case it is not possible to make all numbers one using less than 5 moves.
你有一个长度为 $n$ 的数组 $a$，你可以执行操作。每个操作如下：选择数组 $a$ 中两个相邻的元素，记为 $x$ 和 $y$，并将其中一个替换为 $\gcd(x, y)$，其中 $\gcd$ 表示最大公约数。
请问，最少需要多少次操作，才能使所有元素都变为 $1$？
输入的第一行包含一个整数 $n$（$1 \leq n \leq 2000$）——数组中元素的个数。
第二行包含 $n$ 个用空格分隔的整数 $a_1, a_2, \dots, a_n$（$1 \leq a_i \leq 10^9$）——数组的元素。
如果不可能将所有数变为 $1$，请输出 _-1_；否则，请输出使所有数都等于 $1$ 所需的最少操作次数。
在第一个样例中，你可以使用以下 $5$ 步操作将所有数变为 $1$：
我们可以证明，在这种情况下，不可能用少于 $5$ 步操作使所有数都变为 $1$。
## Input
输入的第一行包含一个整数 $n$（$1 \leq n \leq 2000$）——数组中元素的个数。第二行包含 $n$ 个用空格分隔的整数 $a_1, a_2, \dots, a_n$（$1 \leq a_i \leq 10^9$）——数组的元素。
## Output
如果不可能将所有数变为 $1$，请输出 _-1_；否则，请输出使所有数都等于 $1$ 所需的最少操作次数。
[samples]
## Note
在第一个样例中，你可以使用以下 $5$ 步操作将所有数变为 $1$： $[2, 2, 3, 4, 6]$。 $[2, 1, 3, 4, 6]$ $[2, 1, 3, 1, 6]$ $[2, 1, 1, 1, 6]$ $[1, 1, 1, 1, 6]$ $[1, 1, 1, 1, 1]$ 我们可以证明，在这种情况下，不可能用少于 $5$ 步操作使所有数都变为 $1$。
**Definitions**
Let $ n \in \mathbb{Z} $ be the length of the array.
Let $ A = (a_1, a_2, \dots, a_n) $ be a sequence of positive integers, where $ a_i \in \mathbb{Z}^+ $ for all $ i \in \{1, \dots, n\} $.
**Constraints**
1. $ 1 \leq n \leq 2000 $
2. $ 1 \leq a_i \leq 10^9 $ for all $ i \in \{1, \dots, n\} $
**Objective**
Determine the minimum number of operations required to transform all elements of $ A $ into $ 1 $, where each operation consists of:
- Selecting two adjacent elements $ a_i, a_{i+1} $,
- Replacing either $ a_i $ or $ a_{i+1} $ with $ \gcd(a_i, a_{i+1}) $.
If it is impossible to make all elements equal to $ 1 $, output $ -1 $.
**Key Insight**
It is impossible if and only if $ \gcd(a_1, a_2, \dots, a_n) \neq 1 $.
Otherwise, the minimum number of operations is:
$$
n - \text{(length of the longest contiguous subarray with GCD } = 1) + (\text{number of } 1\text{s already present}) - 1
$$
Equivalently, let $ m $ be the minimum number of adjacent GCD operations required to produce a single $ 1 $ anywhere in the array. Then the total operations needed is:
$$
m + (n - 1)
$$
where $ m $ is the minimal number of operations to create a $ 1 $ from some contiguous subarray.
API Response (JSON)
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