How-many-numbers-between-1-2017-that-aren-t-divisible-by-5-6-7-8-

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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