Mocha and Stars
Description
Mocha wants to be an astrologer. There are $n$ stars which can be seen in Zhijiang, and the brightness of the $i$-th star is $a_i$.
Mocha considers that these $n$ stars form a constellation, and she uses $(a_1,a_2,\ldots,a_n)$ to show its state. A state is called mathematical if all of the following three conditions are satisfied:
- For all $i$ ($1\le i\le n$), $a_i$ is an integer in the range $[l_i, r_i]$.
- $\sum \limits _{i=1} ^ n a_i \le m$.
- $\gcd(a_1,a_2,\ldots,a_n)=1$.
Here, $\gcd(a_1,a_2,\ldots,a_n)$ denotes the greatest common divisor (GCD) of integers $a_1,a_2,\ldots,a_n$.
Mocha is wondering how many different mathematical states of this constellation exist. Because the answer may be large, you must find it modulo $998\,244\,353$.
Two states $(a_1,a_2,\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ are considered different if there exists $i$ ($1\le i\le n$) such that $a_i \ne b_i$.
The first line contains two integers $n$ and $m$ ($2 \le n \le 50$, $1 \le m \le 10^5$) — the number of stars and the upper bound of the sum of the brightness of stars.
Each of the next $n$ lines contains two integers $l_i$ and $r_i$ ($1 \le l_i \le r_i \le m$) — the range of the brightness of the $i$-th star.
Print a single integer — the number of different mathematical states of this constellation, modulo $998\,244\,353$.
Input
The first line contains two integers $n$ and $m$ ($2 \le n \le 50$, $1 \le m \le 10^5$) — the number of stars and the upper bound of the sum of the brightness of stars.
Each of the next $n$ lines contains two integers $l_i$ and $r_i$ ($1 \le l_i \le r_i \le m$) — the range of the brightness of the $i$-th star.
Output
Print a single integer — the number of different mathematical states of this constellation, modulo $998\,244\,353$.
Samples
2 4
1 3
1 2
4
5 10
1 10
1 10
1 10
1 10
1 10
251
5 100
1 94
1 96
1 91
4 96
6 97
47464146
Note
In the first example, there are $4$ different mathematical states of this constellation:
- $a_1=1$, $a_2=1$.
- $a_1=1$, $a_2=2$.
- $a_1=2$, $a_2=1$.
- $a_1=3$, $a_2=1$.