How many numbers from 1 through 60100 are divisible by none of the numbers from 2 through 6? My thoughts on doing this problem:
total numbers is 60100
so from the total I subtract the numbers divisible by 2, 3, 4, 5, and 6.
Yet my answer
60100-30050-20033-15025-12020-10016
is a negative number. How to solve this problem?
 A: We need to consider the primes only. 
The number of Numbers divisible by at least one of $2,3,5$ will be $$\left\lfloor\frac{60100}2\right\rfloor+\left\lfloor\frac{60100}3\right\rfloor+\left\lfloor\frac{60100}5\right\rfloor-\left\lfloor\frac{60100}{2\cdot3}\right\rfloor-\left\lfloor\frac{60100}{2\cdot5}\right\rfloor-\left\lfloor\frac{60100}{3\cdot5}\right\rfloor+\left\lfloor\frac{60100}{2\cdot3\cdot5}\right\rfloor$$
A: you have to add to that negative number the numbers divisible by (2*3), (2*4), (2*5), (2*6),
(3*4), (3*5), (3*6), (4*5), (4*6) and (5*6) as you are subtracting them twice
and subtract again the numbers divisible by (2*3*4),(2*3*5)...(all combinations of 3 numbers)
because you are adding them three times, hence is like if you had not subtracted none of them at the beggining
then add all combinations of 4
and finally subtract (2*3*4*5*6)
A: Firstly, as pointed out by others, you need only eliminate numbers divisible by $2,3,5$
How many numbers between $1$ and $30$ are indivisible by these numbers?
If you add $30$ to such an indivisible number, does its indivisible status change?
How many sets of $30$ numbers are there between $1$ and $60090$?
Can you check the numbers from $60091$ to $60100$ by hand?
