$3780=2^2\cdot3^3\cdot5\cdot7$
Any number that is not co-prime with $3780$ must be divisible by at lease one of $2,3,5,7$
Let us denote $t(n)=$ number of numbers$\le 6042$ divisible by $n$
$t(2)=\left\lfloor\frac{6042}2\right\rfloor=3021$
$t(3)=\left\lfloor\frac{6042}3\right\rfloor=2014$
$t(5)=\left\lfloor\frac{6042}5\right\rfloor=1208$
$t(7)=\left\lfloor\frac{6042}7\right\rfloor=863$
$t(6)=\left\lfloor\frac{6042}6\right\rfloor=1007$
Similarly, $t(30)=\left\lfloor\frac{6042}{30}\right\rfloor=201$
and $t(2\cdot 3\cdot 5\cdot 7)=\left\lfloor\frac{6042}{210}\right\rfloor=28$
The number of number not co-prime with $3780$
=$N=\sum t(i)-\sum t(i\cdot j)+\sum t(i\cdot j \cdot k)-t(i\cdot j\cdot k \cdot l)$ where $i,j,k,l \in (2,3,5,7)$ and no two are equal.
The number of number coprime with $3780$ is $6042-N$
Reference: Venn Diagram for 4 Sets
Nobody's really keeping count.
Newly discovered large primes make the news, but primes in the range of, say, a few hundred digits are not something that anybody keeps track of. They are very easy to find -- the computer that's showing you this text is likely capable of finding at least several ones per second for you, and with overwhelming probability they will be primes nobody else have ever seen before.
There are very many hundred-digit primes to find. We could cover the Earth in harddisks full of distinct hundred-digit primes to a height of hundreds of meters, without even making a dent in the supply of hundred-digit primes.
This also raises the question of what it means that a prime is "known". If I generate a dozen hundred-digit primes and they are forgotten after I close the window showing them, are these primes still "known"? If instead I print out one of them and save the copy in a safe without showing it to anybody, is that prime "known"? What if I cast it into the concrete foundation for my new house?
Best Answer
Define ''the amount of primes numbers below $n$'' as $\pi(n)$, the prime-counting function. Then the question ''How many prime numbers are there from 43 to 60?'' can be ''translated'' as $$ \pi(60) - \pi(43), $$ which is equal to $3$.