Mocha and Stars

2000ms

262144K

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 $$$a_i \ne b_i$$$.

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 $$$998\,244\,353$$$.

Sample Input:

2 4
1 3
1 2

Sample Output:

Sample Input:

Sample Output:

Sample Input:

Sample Output:

47464146

Note:

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

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

Informação

Provedor Codeforces

Código CF1559E