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


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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

	$2, 3 a n d$ $m . c . m (2, 3 a n d 5) = 30$ $D_{30 =} ⌊ 900 / 30 ⌋ = 30$ $2, 3 o r 5$ $D_{2} = ⌊ 900 / 2 ⌋ = 450$ $D_{3} = ⌊ 900 / 3 ⌋ = 300$ $D_{5} = ⌊ 900 / 5 ⌋ = 180$ $D_{6} = ⌊ 900 / 6 ⌋ = 150$ $D_{10} = . . . = 90$ $D_{15} = 80$ $D_{30} = 30$ $D_{2, 3 o r 5} = 450 + 300 + 180 - 150 - 90 - 80 + 30 =$ $= 640$
	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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