Question

find the number of integers between 1 and 250 that are divisible by any of the integers 2,3,5 and 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.

Explanation:

Answer 2

Answer:

There are 193 integers between 1 and 250 both inclusive that are divisible by 2,3,5 and 7.  

Explanation:

Let A set of integers that are divisible by 2.

     B set of integers that are divisible by 3.

     C set of integers that are divisible by 5.

     D set of integers that are divisible by 7.

IA∪B∪C∪Dl ≈ IAI + IBI + ICI + IDI - IA∩BI - IA∩CI - IA∩DI - IB∩CI - IB∩DI - IC∩DI +IA∩B∩CI + IA∩B∩DI + IB∩C∩DI + IC∩A∩DI - IA∩B∩C∩DI

• IAI ≈ [250÷2] ≈ 125

• IBI = [205÷3] = 83

• ICI ≈ [250÷5] = 50

• IDI = [250÷7] ≈ 35

• IA∩BI = [250÷2×3] ≈ 41

• IA∩CI = [250÷2×5] = 25

• IA∩DI = [250÷2×7] = 17

• IB∩CI = [250÷3×5] = 16

• IB∩DI ≈ [250÷3×7] = 11

• IC∩DI = [250÷5×7] = 7

• IA∩B∩CI = [250÷2×3×5] = 8

• IA∩B∩DI = [250÷2×3×7] = 5

• IB∩C∩DI = [250÷3×5×7] = 2

• IC∩A∩DI = [250÷5×2×7] = 3

• IA∩B∩C∩DI =  [250÷2×3×5×7] = 1

Now substitute the values obtained in the formula

IA∪B∪C∪DI =  125+83+50+35-41-25-17-16-11-7+8+5+2+3-1 = 193.

∴ in between 1 and 250, there are 193 integers including 1 and 250 that are divisible by 2,3,5 and 7.