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Algebra

Blazing goIITian

 Joined: 2 Mar 2012 Post: 308
22 Mar 2012 18:56:55 IST
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Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

If N is a prime number which divides S then the largest possible value of N among following is (a) 41 (b) 31 (c) 37 (d) 19

Blazing goIITian

Joined: 2 Mar 2012
Posts: 308
22 Mar 2012 18:58:52 IST
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## S = 39P19 + 38P19 + 37P19 + .............. ...... + 20P19 .

Cool goIITian

Joined: 21 Jul 2010
Posts: 88
22 Mar 2012 19:24:36 IST
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the above expression can be reduced to

## S = 39P19 + 38P19 + 37P19 + .............. ...... + 20P19 . =  19!(40C20 -1)

so i think (d)19 should be the ans.

Blazing goIITian

Joined: 2 Mar 2012
Posts: 308
22 Mar 2012 19:45:22 IST
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41 is given answer . pls justify.

Cool goIITian

Joined: 21 Jul 2010
Posts: 88
22 Mar 2012 20:05:15 IST
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ahh yes its 41.

19!(40C20 -1) can be further written as 19![(40!)/20!20! -1]

= 19! [ (40*39*38*37......*21)/20! -1]

= 2*39*38*37....21 - 19!

= 2*(41-2)(41-3)(41-4).....(41-20) - 19!

= 41a - 2*2*3*4*...20 - 19!

= 41a - 2*20!-19!

= 41a -19!(40 +1)

= 41(a-19!)

so its clearly divisible by 41

Blazing goIITian

Joined: 2 Mar 2012
Posts: 308
22 Mar 2012 20:19:59 IST
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gr8, manh!! fully agreed !! perfect !!

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