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8 Mar 2010 00:08:50 IST

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prove that 31^3 | 100!

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prove that 31^3 | 100!

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NugoRama

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Joined: 11 Mar 2009

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8 Mar 2010 14:22:05 IST

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Assuming the question to be...

Prove that 31^3 divides 100!.

100! = 1x2x3x...x31x....x62x.....x93x....100.

= (something natural) x (31x62x93)

=(something bigger and natural) x (31x31x31)

hence, 31^3 divides 100!.

The end..

edison

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Joined: 19 Oct 2006

Posts: 7449

31 Jan 2011 17:56:15 IST

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100 ! can be expressed as

100! = 1.2.3.....31....62.....93....100

or 100 ! = k . 31 . 62 . 93 (where k is some natural number)

Thus 100 ! / 31³ = k . 31 . 62 . 93 / 31³

or 100 ! / 31³ = 1.2.3.k . 31 . 31 . 31 / 31³

or 100 ! / 31³ = 6k (which is an integer)

Thus 100! is divisible by 31³

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