Let S be the set of integers x such that
I. 100 < x < 200,
II x is odd
III. x is divisible by 3 but not by 7?
How many elements does S contain?

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Solution

The correct option is D 13
$T h e g i v e n r a n g e s i s 100 < x < 200. S o, t h e n u m b e r s a r e - 101, 102, 103, - - - - - 199. O u t o f t h e m t h e f i r s t o d d n u m b e r d i v i s i b l e b y 3 i s 105 a n d l a s t s u c h n u m b e r i s 195. ∵ W e e x c l u d e t h e e v e n m u l t i p l e s o f 3 t h e n e a c h n u m b e r c a n b e w r i t t e n b y a d d i n g 6 t o t h e p r e v i o u s n u m b e r . ∴ N o w t h e n u m b e r s a r e 105, 111, 117, - - - - -, 195. T h i s i s a n A . P s e r i e s w i t h c o m m o n d i f f e r e n c e 6. A p p l y i n g t h e r u l e n = \frac{l - a}{b} + 1 w e g e t t h e n u m b e r o f t e r m s i s n = \frac{195 - 105}{6} + 1 = 16 - - - - (1) N o w t h e n u m b e r s c a n b e w r i t t e n a s 3 \times 35, 3 \times 37, 3 \times 39 - - - -, 3 \times 65 S o t h e m u l t i p l i e r s o f 3 a r e, i n e a c h n u m b e r 35, 37, 39, - - - -, 63, 65. O u t o f t h e s e w h i c h h a v e 7 a s a f a c t o r a r e 7 \times 5, 7 \times 7 a n d 7 \times 9. T o t a l 3 - - - - (2) ∴ T o t a l n u m b e r o f n u m b e r s w h i c h c o m p l i e s w i t h a l l t h e g i v e n c o n d i t i o n i s 16 - 3 = 13 [f r o m (1) a n d (2)] ∴ T h e s e t s h a s 13 e l e m e n t s .$

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