20070809, 17:53  #1 
May 2004
New York City
3·17·83 Posts 
Divisible by 7 ?
Show that C(1000,500) = _{1000}C_{500} = (1000!)/(500!)^{2}
is in fact NOT divisible by 7. (Without of course multiplying it out.) 
20070809, 18:52  #2  
Nov 2003
2^{2}×5×373 Posts 
Quote:
floor(1000/7) + floor(1000/49) + floor(1000/343) = 162 Counting the number of time 500! is divisible by 7 yields (similarly) 81. Thus (500!)^2 is divisible by 7 162 times, as is 1000!. QED 

20070809, 19:22  #3  
May 2004
New York City
4233_{10} Posts 
Quote:
142 + 20 + 2 = 164, and the sum for 500 to be 71 + 10 + 1 = 82, which gives the same result about divisibility. 

20070809, 20:02  #4 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
2^{2}·5·233 Posts 
Didn't he just prove it IS DIVISIBLE BY 7?

20070809, 20:10  #5 
Aug 2002
Buenos Aires, Argentina
2^{4}·5·17 Posts 
He proved that it is not divisible by 7.
Let a = 1000! = k*7^164 and b = 500!^2 = m*7^164 (where k and m are not divisible by 7). So we get a/b = k/m. The fraction a/b is not divisible by 7 because k is not divisible by 7. 
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