4. How many natural numbers are there that are less than or equal to the number 2019, and which are not divisible by any of the numbers 6, 9, and 15?
First method:

To count how many natural numbers are less than or equal to 2019 and are not divisible by given numbers, we count all the numbers that are divisible by these numbers.

The numbers divisible by 6 are 336, because \(2019 \div 6 = 336\) with a remainder of 3.

The numbers divisible by 9 are 224, because \(2019 \div 9 = 224\) with a remainder of 3.

The numbers divisible by 15 are 134, because \(2019 \div 15 = 134\) with a remainder of 9.

If we sum the multiples of each individual number, some of them have been counted multiple times.

The numbers divisible by both 6 and 9, i.e., by \(\text{LCM}(6,9) = 18\), have been counted twice.

Similarly, the numbers divisible by \(\text{LCM}(6,15) = 30\) and \(\text{LCM}(9,15) = 45\) have been counted twice.

The numbers divisible by \(\text{LCM}(6,9,15) = 90\) have been counted three times.

The numbers divisible by 18 are 112, because \(2019 \div 18 = 112\) with a remainder of 3.

The numbers divisible by 30 are 67, because \(2019 \div 30 = 67\) with a remainder of 9.

The numbers divisible by 45 are 44, because \(2019 \div 45 = 44\) with a remainder of 39.

The numbers divisible by 90 are 22, because \(2019 \div 90 = 22\) with a remainder of 39.

\[336 + 224 + 134 - (112 + 67 + 44) + 22 = 694 - 223 + 22 = 493\]

There are 493 natural numbers less than or equal to 2019 that are divisible by 6, 9, or 15.

The desired numbers then have \(2019 - 493 = 1526\).

There are 1526 natural numbers less than or equal to 2019 that are not divisible by any of the numbers 6, 9, and 15.