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Method of Finding HCF

The number of...

Question

The number of positive integers n in the range $12 \leq n \leq 40$ such that the product (n - 1) (n - 2) . . . 2. 1 is not divisible by n is:

A

5

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B

7

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C

13

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D

14

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Solution

The correct option is D 7
$T h e g i v e n e x p r e s s i o n i s - f (x) = (n - 1) (n - 2) (n - 3) - - - - 2 (a) S i n c e t h e l a s t f a c t o r i s 2 t h e p r o d u c t i s d i v i s i b l e b y a t l e a s t 2 o r a n e v e n n u m b e r . (b) E v e n i f n i s a c o m p o s i t o d d n u m b e r f (x) c a n c o n t a i n f a c t o r s w h i c h a r e f a c t o r s o f n a l s o . S o n c a n n o t b e a c o m p o s i t n u m b e r . (c) L e t u s t a k e n a s a p r i m e . C l e a r l y f (x) i s a p r o d u c t o f s u c c e s s s i v e n u m b e r t a k e n i n d e s c e n d i n g o r d e r . T h e r e f o r e (n - 1) (n - 2) - - - - 2 d o e s n o t h a v e a n y f a c t o r w h i c h i s d i v i s i b l e b y n . (s i n c e n i s a p r i m e i t h a s n o o t h e r f a c t o r t h a n n a n d 1 a n d t h e p r o d u c t d o e s n o t c o n t a i n n) . T h e r e f o r e n i s a p r i m e n u m b e r . N o w t h e r a n g e i s 12 \leq n \leq 40. T h e r e f o r e t h e p o s s i b l e n u m b e r s a r e 12, 13, 14 - - - - 40. T h e r e a r e 7 p r i m e s i n t h i s l i s t . ∴ 7 i s t h e a n s w e r .$

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