Linear Congruential Generator

Linear Congruential Generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. (source: wikipedia)
Anyway, here's the fomula:

function f(Si) {
  return (a * Si + b) mod N
}

In short, a, b, and N are the coefficients of the random number generator (RNG).
So in round 1, S1 will be plugged in, and produce the output S2. Then in the next round, S2 will become the input and produce S3, so on and so forth. By gathering all the outputs, we got a sequence of random numbers!

The question is, can we crack this RNG? That is, can we determine the coefficients (a, b, and N) by simply observing the output sequence? Let's check it out!

Playground

a = , b = , N =

Calculate (2 * + 7) mod 23:

Or calculate anything:

1. We already leaked two of them, a = 12349876 and N = 43216789 , please help us figure out b!

Here are some outputs we've gathered:

Si   = 34425850
Si+1 = 19828421
Si+2 = 40149722

Once we figured out b, ALL the consecutive numbers generated would no longer be random anymore (aka cracked)!
If you think you've cracked it, please provide the next random number:

Si+3 =

2. This time, we only leaked N = 87879487 , but I somehow feel that it's still crackable...

Again, we have gathered some outputs:

Si   = 30296335
Si+1 = 60248031
Si+2 = 23522917

If you think you've cracked it, please provide the next random number:

Si+3 =

3. This time we had no luck and leaked none of them. To compensate that, we've gathered more outputs, which might help in cracking it :-)

Anyway, here are the outputs we've gathered:

Si   = 67405306
Si+1 = 38737998
Si+2 = 21870365
Si+3 = 72754763
Si+4 = 32045724
Si+5 = 48184566
Si+6 = 54253663

If you think you've cracked it, please provide the next random number:

Si+7 =

Hint: Crack N first by using the fact that the probability of k randomly chosen integers being coprime is 1/zeta(k).