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Prime Factor (Posted on 2006-05-12) Difficulty: 4 of 5
Determine the largest 3 - digit prime factor of 2000 C 1000.

n C r denotes the number of combinations of n things taking r at a time.

No Solution Yet Submitted by Ravi Raja    
Rating: 2.5000 (2 votes)

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Solution Not quite the beast | Comment 5 of 8 |

The largest 3-digit number is 661.

2000C1000 = 2000!/1000!(2000-1000)! = 2000!/1000!(1000!)

=2000*1999*1998*....*1003*1002*1001/1000*999*998*....*3*2*1

=(1000*2)*1999*(999*2)*1997*....*(501*2)*1001/1000*....

= 2*1999*2*1997*..../500*499...

Leaving only odd numbers between 2000 and 1001 in the numerator and no number greater than 500 in the denominator.  |_ 2000/3 _| = 666. The largest prime between 500 and 666 is 661 (= 1983/3).   Therefore 661 is the largest 3-digit prime factor of 2000C1000.


  Posted by Dej Mar on 2006-05-12 14:51:30
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