A14561. Mocha and Stars

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)](https://en.wikipedia.org/wiki/Greatest_common_divisor) 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 $998\,244\,353$ .

In the first example, there are $4$ different mathematical states of this constellation:

- $a_1=1$ , $a_2=1$ .
- $a_1=1$ , $a_2=2$ .
- $a_1=2$ , $a_2=1$ .
- $a_1=3$ , $a_2=1$ .