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