codeforces#P1209A. Paint the Numbers
Paint the Numbers
Description
You are given a sequence of integers $a_1, a_2, \dots, a_n$. You need to paint elements in colors, so that:
- If we consider any color, all elements of this color must be divisible by the minimal element of this color.
- The number of used colors must be minimized.
For example, it's fine to paint elements $[40, 10, 60]$ in a single color, because they are all divisible by $10$. You can use any color an arbitrary amount of times (in particular, it is allowed to use a color only once). The elements painted in one color do not need to be consecutive.
For example, if $a=[6, 2, 3, 4, 12]$ then two colors are required: let's paint $6$, $3$ and $12$ in the first color ($6$, $3$ and $12$ are divisible by $3$) and paint $2$ and $4$ in the second color ($2$ and $4$ are divisible by $2$). For example, if $a=[10, 7, 15]$ then $3$ colors are required (we can simply paint each element in an unique color).
The first line contains an integer $n$ ($1 \le n \le 100$), where $n$ is the length of the given sequence.
The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 100$). These numbers can contain duplicates.
Print the minimal number of colors to paint all the given numbers in a valid way.
Input
The first line contains an integer $n$ ($1 \le n \le 100$), where $n$ is the length of the given sequence.
The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 100$). These numbers can contain duplicates.
Output
Print the minimal number of colors to paint all the given numbers in a valid way.
Samples
6
10 2 3 5 4 2
3
4
100 100 100 100
1
8
7 6 5 4 3 2 2 3
4
Note
In the first example, one possible way to paint the elements in $3$ colors is:
- paint in the first color the elements: $a_1=10$ and $a_4=5$,
- paint in the second color the element $a_3=3$,
- paint in the third color the elements: $a_2=2$, $a_5=4$ and $a_6=2$.
In the second example, you can use one color to paint all the elements.
In the third example, one possible way to paint the elements in $4$ colors is:
- paint in the first color the elements: $a_4=4$, $a_6=2$ and $a_7=2$,
- paint in the second color the elements: $a_2=6$, $a_5=3$ and $a_8=3$,
- paint in the third color the element $a_3=5$,
- paint in the fourth color the element $a_1=7$.