Using the principle of inclusion-exclusion, the number of integers between 1 and 2000 (inclusive) that are divisible by at least one of 2, 3, 5, 7 is

For all integers x ∈ [1,2000]
N(2|x)=2000/2=1000
N(3|x)=2000/3=666
N(5|x)=400
N(7|x)=285
Total =2351

N(2|x & 3|x)=N(6|x)=333
N(15|x)=133
N(35|x)=57
N(10|x)=200
N(14|x)=142
N(21|x)=95
total = 960

N(2|x & 3|x & 5|x)=N(30|x)=66
N(42|x)=47
N(70|x)=28
N(105|x)=19
total=160

N(2|x & 3|x & 5|x & 7|x)=N(210|x)=9
total=9

By the inclusion/exclusion principle, numbers of integers between 1 and 2000 divisible by at least one of 2,3,5,7
= 2351-960+160-9
=1542