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# 1625. Lexicographically Smallest String After Applying Operations

<https://leetcode.com/problems/lexicographically-smallest-string-after-applying-operations>

## Description

You are given a string `s` of **even length** consisting of digits from `0` to `9`, and two integers `a` and `b`.

You can apply either of the following two operations any number of times and in any order on `s`:

* Add `a` to all odd indices of `s` **(0-indexed)**. Digits post `9` are cycled back to `0`. For example, if `s = "3456"` and `a = 5`, `s` becomes `"3951"`.
* Rotate `s` to the right by `b` positions. For example, if `s = "3456"` and `b = 1`, `s` becomes `"6345"`.

Return *the **lexicographically smallest** string you can obtain by applying the above operations any number of times on* `s`.

A string `a` is lexicographically smaller than a string `b` (of the same length) if in the first position where `a` and `b` differ, string `a` has a letter that appears earlier in the alphabet than the corresponding letter in `b`. For example, `"0158"` is lexicographically smaller than `"0190"` because the first position they differ is at the third letter, and `'5'` comes before `'9'`.

**Example 1:**

```

**Input:** s = "5525", a = 9, b = 2
**Output:** "2050"
**Explanation:** We can apply the following operations:
Start:  "5525"
Rotate: "2555"
Add:    "2454"
Add:    "2353"
Rotate: "5323"
Add:    "5222"
​​​​​​​Add:    "5121"
​​​​​​​Rotate: "2151"
​​​​​​​Add:    "2050"​​​​​​​​​​​​
There is no way to obtain a string that is lexicographically smaller then "2050".
```

**Example 2:**

```

**Input:** s = "74", a = 5, b = 1
**Output:** "24"
**Explanation:** We can apply the following operations:
Start:  "74"
Rotate: "47"
​​​​​​​Add:    "42"
​​​​​​​Rotate: "24"​​​​​​​​​​​​
There is no way to obtain a string that is lexicographically smaller then "24".
```

**Example 3:**

```

**Input:** s = "0011", a = 4, b = 2
**Output:** "0011"
**Explanation:** There are no sequence of operations that will give us a lexicographically smaller string than "0011".
```

**Example 4:**

```

**Input:** s = "43987654", a = 7, b = 3
**Output:** "00553311"
```

**Constraints:**

* `2 <= s.length <= 100`
* `s.length` is even.
* `s` consists of digits from `0` to `9` only.
* `1 <= a <= 9`
* `1 <= b <= s.length - 1`

## ac

```java
```