Answer to Question #137265 in Discrete Mathematics for Pri

Question

Among the integer 1 to 300,find how many are not divisible by 3,nor by 5 also find how many are divisible by 3but not by 7

Answer 1

let D={1,...,300} - all integers from 1 to 300,

A={3k; k=1,...,100} - integers divisible by 3 from 1 to 300,

B={5k; k=1,...,60} - integers divisible by 5 from 1 to 300,

C={7k; k=1,...,42} - integers divisible by 7 from 1 to 300.

1) we need to find |D\(A∪B)| i.e cardinality of the set of number from 1 to 300 except divisible by 3 and 5.

|A∪B|=|A|+|B|-|A∩B|

A∩B={15k; k=1,...,20} - integers divisible by both 3 and 5 from 1 to 300. Then

|A∪B|=100+60-20=140. Then

|D\(A∪B)|=|D|-|A∪B|=300-140=160.

Answer to question 1) is 160.

2)we need to find |A\C|. This is the same as |A\(A∩C)| i.e cardinality of the set of number from 1 to 300 divisible by 3 but not by 7.

A∩C={21k; k=1,...,14} - integers divisible by both 3 and 7 from 1 to 300. Then

|A\(A∩C)|=|A|-|A∩C|=100-14=86.

Answer to question 2) is 86.