Question

4. Find the number of integers between 1 and 250 both inclusive that are divisible by any of the integers 2, 3,5,7

Answer 1

Answer:

SOLUTION: the number of integers between 1 and 250 that are divisible by2,5,7 is. I'll denote {x|n} be the set of integers x <= 250 that divide n. ... Hence, the number of integers is 125 + 50 + 35 - 25 - 17 - 7 + 3 = 164.

Answer 2

Answer:

Formula

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

Calculation:

Given 1 ≤ n ≤ 250

Let

A: Integers divisible by 2

B: Integers divisible by 3

C: Integers divisible by 7

Therefore,

n(A) = number divisible by 2 =

n(B) = number divisible by 3 =

n(C) = number divisible by 7 =

n(A ∩ B) = number divisible by both 2 and 3 (i.e. 6)

n(B ∩ C) = number divisible by both 3 and 7 (i.e. 21) =

n(C ∩ A) = number divisible by both 7 and 2 (i.e. 14)

n(A ∩ B ∩ C) = number divisible by 2, 3 and 7 (i.e. 42) =

By using the above formula,

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

Number of integers between 1 and 250, that are divisible by any of the integer 2, 3 and 7 will be,

n(A ∪ B ∪ C) = 125 + 83 + 35 - 41 - 11 - 17 + 5

n(A ∪ B ∪ C) = 179