Among integers from 1 to 1000 how many numbers are not divisible by 3 nor by 5 nor by 7 ........

Mathematics (NDA) Set theory

Among integers from 1 to 1000, how many numbers are not divisible by 3 nor by 5 nor by 7?

500

645

457

784

Let A set of numbers between 1 to 1000 that are divisible by 3

B set of numbers between 1 to 1000 that are divisible by 5

C set of numbers 1 to 1000 that are divisible by 7

|A|=|1000/3|=333

|B|=|1000/5|=200

And |C|=|1000/7|=142

|A?B?C|=|1000/3 × 5 × 7 |=9

|A?B|=|1000/3× 5|=66

|B?C|=|1000/5× 7|=28

|A?C|=|1000/3×7|=47

|U|-|A ?B?C| not divisible by 3,5,7

|A ?B?C|=|A|+|B|+|C|-|A?B|-|A?C|-|B?C|+|A?B?C|

=333+200+142-66-47-28+9

=543

|U|-|A ?B?C| = 1000-543=457

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