How many numbers between 1 − 2017 that aren′t divisible by 5, 6, 7, 8 ?


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Question Number 11754 by Joel576 last updated on 31/Mar/17

$How many numbers between 1 - 2017$ $that {aren}^{'} t divisible by 5, 6, 7, 8 ?$
	Answered by mrW1 last updated on 31/Mar/17

	$l e t a = c o u n t o f n u m b e r s w h i c h a r e m u l t i p l e o f 5$ $l e t b = c o u n t o f n u m b e r s w h i c h a r e m u l t i p l e o f 6$ $l e t c = c o u n t o f n u m b e r s w h i c h a r e m u l t i p l e o f 7$ $l e t d = c o u n t o f n u m b e r s w h i c h a r e m u l t i p l e o f 8$ $a = 2017 / 5 = 403$ $a_{1} = 2017 / (5 \times 6) = 67 (a l s o i n b)$ $a_{2} = 2017 / (5 \times 7) = 57 (a l s o i n c)$ $a_{3} = 2017 / (5 \times 8) = 50 (a l s o i n d)$ $b = 2017 / 6 = 336$ $b_{1} = 2017 / (6 \times 7) = 48 (a l s o i n c)$ $b_{2} = 2017 / (6 \times 8) = 42 (a l s o i n d)$ $c = 2017 / 7 = 288$ $c_{1} = 2017 / (7 \times 8) = 36 (a l s o i n d)$ $d = 2017 / 8 = 252$ $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, c_{1} a r e c o u n t o f n u m b e r s$ $w h i c h a r e d o u b l e c o u n t e d .$ $c o u n t o f n u m b e r s w h i c h a r e d i v i s i b l e$ $b y 5, 6, 7, 8 :$ $a + b + c + d - a_{1} - a_{2} - a_{3} - b_{1} - b_{2} - c_{1}$ $= 403 + 336 + 288 + 252 - 67 - 57 - 50 - 48 - 42 - 36$ $= 979$ $c o u n t o f n u m b e r s w h i c h a r e n o t d i v i s i b l e$ $b y 5, 6, 7, 8 :$ $2017 - 979 = 1038$
	Commented by Joel576 last updated on 01/Apr/17

	$t h a n k y o u v e r y m u c h$
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