Right now, count feels like an arbitrary number, and it may seem like we're done. However, there is more! We need to consider all the different N that have some common multiples.

For example, if one N is 5, we count all of its multiples: 5, 10, 15, 20...

If another N is 3, we count: 3, 6, 9, 12, 15, 18...

This means that we're overcounting certain numbers, since they're multiples of different N. And we're overcounting the multiples of these certain numbers too. In this example, the number 30 is also counted in both lists. So is 45. This means that we need to find a list of the least common multiples between various N and we need to subtract them and their multiples from our count .

Computing all the LCMs for so many numbers however is going to be very computationally expensive again. A naive approach I tried ran for 30 mins straight without yielding any answer at all. So I started looking for patterns, and I found one.

It seems like the LCMs of all the combinations of prime multiples formed by choosing 4 primes out of 5 are the same. That sentence is a bit dense, so let's unpack it.

You have 5 primes as follows, and you create all the possible prime multiples you can by choosing 4 out of the 5.