Find-the-least-positive-integer-n-for-which-there-exists-a-set-s-1-s-2-s-3-s-n-consisting-of-n-distinct-positive-integers-such-that-1-1-s-1-1-1-s-2-1-1-s-3-1-1-s-n-

Question Number 103908 by bobhans last updated on 18/Jul/20

F i n d t h e l e a s t p o s i t i v e i n t e g e r n f o r

w h i c h t h e r e e x i s t s a s e t {s_{1}, s_{2}, s_{3}, \dots, s_{n}}

c o n s i s t i n g o f n d i s t i n c t p o s i t i v e i n t e g e r s

s u c h t h a t (1 - \frac{1}{s_{1}}) (1 - \frac{1}{s_{2}}) (1 - \frac{1}{s_{3}}) \dots (1 - \frac{1}{s_{n}})

= \frac{51}{2010} .

Commented by Rasheed.Sindhi last updated on 18/Jul/20

\frac{51}{2010} = \frac{17}{670}

Commented by bobhans last updated on 18/Jul/20

f o r n = ? ? s i r

Answered by mr W last updated on 18/Jul/20

i g o t :

n_{m i n} = 9

s_{1} = 67

s_{2} = 25

s_{3} = 18

s_{4} = 11

s_{5} = s_{6} = s_{7} = s_{8} = s_{9} = 2

b u t t h e y a r e n o t d i s t i n c t!

Answered by john santu last updated on 18/Jul/20

s u p p o s e t h a t f o r n t h e r e e x i s t

t h e d e s i r e d n u m b e r s; w e m a y

a s s u m s e t h a t s_{1} < s_{2} < s_{3} < \dots < s_{n} .

s u r e l y s_{1} > 1 s i n c e o t h e r w i s e

1 - \frac{1}{s_{1}} = 0 . s o w e h a v e 2 ⩽ s_{1} ⩽ s_{2} - 1 ⩽ \dots ⩽ s_{n} - (n - 1),

h e n c e s_{i} ⩾ i + 1 f o r e a c h

i = 1, 2, \dots, n . T h e r e f o r e \frac{51}{2010} =

(1 - \frac{1}{s_{1}}) (1 - \frac{1}{s_{2}}) \dots (1 - \frac{1}{s_{n}}) ⩾ (1 - \frac{1}{2}) (1 - \frac{1}{3}) \dots (1 - \frac{1}{n + 1}) = \frac{1}{2} . \frac{2}{3} . \frac{3}{4} \dots \frac{n}{n + 2}

w h i c h i m p l i e s n + 1 ⩾ \frac{2010}{51} = \frac{670}{17} > 39

s o n ⩾ 39 . N o w w e a r e l e f t t o

s h o w t h a t n = 39 f i t s . c o n s i d e r

t h e s e t {2, 3, 4, \dots 33, 34, \dots, 67}

w h i c h c o n t a i n s e x a c t l y 39

n u m b e r s . w e h a v e \frac{1}{2} . \frac{2}{3} \dots \frac{32}{33} . \frac{33}{34} \dots \frac{39}{40} . \frac{66}{67}

= \frac{1}{33} . \frac{34}{40} . \frac{66}{67} = \frac{17}{670} = \frac{51}{2010} .

h e n c e f o r n = 39 t h e r e e x i s t a

d e s i r e d e x a m p l e . (J S ⊛)

Commented by bobhans last updated on 18/Jul/20

t h a n k y o u

Commented by mr W last updated on 18/Jul/20

v e r y n i c e!

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