How-many-natural-number-from-1-to-900-which-are-not-divisible-2-3-and-5-

Question Number 155856 by peter frank last updated on 05/Oct/21

How many natural number from

1 to 900 which are not divisible

2, 3 and 5

Answered by TheSupreme last updated on 05/Oct/21

2, 3 a n d

m . c . m (2, 3 a n d 5) = 30

D_{30 =} ⌊ 900 / 30 ⌋ = 30

2, 3 o r 5

D_{2} = ⌊ 900 / 2 ⌋ = 450

D_{3} = ⌊ 900 / 3 ⌋ = 300

D_{5} = ⌊ 900 / 5 ⌋ = 180

D_{6} = ⌊ 900 / 6 ⌋ = 150

D_{10} = \dots = 90

D_{15} = 80

D_{30} = 30

D_{2, 3 o r 5} = 450 + 300 + 180 - 150 - 90 - 80 + 30 =

= 640

Commented by puissant last updated on 05/Oct/21

S i r y o u u s e w h i c h f o r m u l a . ?

Answered by mr W last updated on 05/Oct/21

n u m b e r s d i v i s i b l e b y 30 :

⌊ \frac{900}{30} ⌋ = 30

n u m b e r s d i v i s i b l e b y 6 :

⌊ \frac{900}{6} ⌋ - 30 = 120

n u m b e r s d i v i s i b l e b y 10 :

⌊ \frac{900}{10} ⌋ - 30 = 60

n u m b e r s d i v i s i b l e b y 15 :

⌊ \frac{900}{15} ⌋ - 30 = 30

n u m b e r s d i v i s i b l e b y 2 :

⌊ \frac{900}{2} ⌋ - 30 - 120 - 60 = 240

n u m b e r s d i v i s i b l e b y 3 :

⌊ \frac{900}{3} ⌋ - 30 - 120 - 30 = 120

n u m b e r s d i v i s i b l e b y 5 :

⌊ \frac{900}{5} ⌋ - 30 - 60 - 30 = 60

t o t a l n u m b e r s d i v i s i b l e b y 2, 3, 5 :

30 + 120 + 60 + 30 + 240 + 120 + 60 = 660

n u m b e r s n o t d i v i s i b l e b y 2, 3, 5 :

900 - 660 = 240

Commented by mr W last updated on 05/Oct/21

⌊ \frac{900}{7} ⌋ = 128 m e a n s t h e r e a r e 128

n u m b e r s f r o m 1 t o 900 w h i c h a r e

d i v i s i b l e b y 7 .

Commented by puissant last updated on 05/Oct/21

S i r y o u u s e w h i c h f o r m u l a . . ?

Commented by mr W last updated on 05/Oct/21

i u s e d n o s p e c i a l f o r m u l a .

Commented by puissant last updated on 05/Oct/21

T h a n k s s i r . .

Commented by peter frank last updated on 06/Oct/21

which is correct 640 or 240 ?

Commented by mr W last updated on 06/Oct/21

t r y t o u n d e r s t a n d t h e s o l u t i o n, t h e n

y o u c a n k n o w w h a t i s t h e c o r r e c t

a n s w e r .

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