Given $n = 1 + x$ and x is the product of four consecutive integers. Then which of the following is true?

n is an odd numbers

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n is prime

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sometimes a perfect square

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All the given

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Solution

The correct options are
A n is an odd numbers
C sometimes a perfect square
$L e t x = a (a + 1) (a + 2) (a + 3) N o w w h e t h e r a i s e v e n o r o d d, t h e f o u r c o n s e c u t i v e n u m b e r^{'} s p r o d u c t c o n t a i n 2, 3, 4 a s f a c t o r s . ∴ x i s d i v i s i b l e b y 2 \times 3 \times 4 = 24 - - - - (1) O p t i o n A ⟶ F o u r c o n s e c u t i v e n u m b e r s c o n t a i n t w o e v e n n u m b e r s . T h e r e f o r e p r o d u c t i s e v e n . ∴ x + 1 s h o u l d b e o d d . O p t i o n A i s c o r r e c t . O p t i o n B ⟶ x i s d i v i s i b l e b y 24 (f r o m 1) N o w t h e r e e x i s t s n o p r i m e o f t h e f o r m 24 p + 1 w h e r e p i s a n a t u r a l n u m b e r . e . g . 29 = 24 \times 1 + 5 31 = 24 \times 1 + 7 37 = 24 \times 1 + 13 W i t h i n c r e a s i n g c o u n t t h e s e c o n d t e r m i n c r e a s e s b e c a u s e d i f f e r e n c e b e t w e e n p r i m e s i n c r e a s e w i t h h i g h e r c o u n t . T h e r e f o r e x + 1 = n i s n o t a p r i m e u n d e r t h e g i v e n c o n d i t i o n . F o r a e x p a n d e d n u m b e r t o b e s q u a r e, t h e f i r s t a n d l a s t t e r m s h o u l d b e s q u a r e n u m b e r . O p t i o n C ⟶ x = a (a + 1) (a + 2) (a + 3) w h e n a i s e v e n a = 2 p ∴ x = 2 p (2 a + 1) (2 a + 2) (2 a + 3) = 4 p (2 p + 1) (p + 1) (2 p + 3) T h e p r o d u c t i s 4 \times 1 \times 1 \times 3 = 12 I f w e a d d 1 t h e n l a s t t e r m o f t h e p r o d u c t i s 13 w h i c h i s n o t a s q u a r e n u m b e r . S o x + 1 = n i s n o t a s q u a r e n u m b e r w h e n a i s e v e n . (2) W h e n a i s o d d - x = (2 p + 1) (2 p + 2) (2 p + 3) (2 p + 4) T h e l a s t t e r m o f t h e p r o d u c t i s 1 \times 2 \times 3 \times 4 = 24. 24 + 1 = 25 i s a s q u a r e t e r m . ∴ n = x + 1 i s a s q u a r e n u m b e r w h e n a i s o d d . ∴ O p t i o n C i s c o r r e c t w h e n a i s o d d . O p t i o n D ⟶ O b v i o u s l y o p t i o n D i s n o t c o r r e c t .$

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n = 1 + x

x

(a + b + c + d + e) (v + w + x + y + z) = N,

v = a + 1

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