find the number of integers between 1 and 250 that are divisible by any of the integers 2,3,5 and 7.
Answers
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:
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.