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Constant and Ratio Puzzle (Posted on 2016-01-30) Difficulty: 3 of 5
Determine all possible values of a positive integer constant C for which there exists an infinite number of pairs (A, B) of positive integers such that:
(A+B-C)!/(A!*B!) is an integer.

No Solution Yet Submitted by K Sengupta    
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quite general Comment 1 of 1
As posted the puzzle seems quite general. 

With B>C it's always posible to adjust A to have an integer for every B value. If this is true the answer would be that for every value of C there exists an infinite number of pairs....

For example: B>C, B-C=D the expression is (A+D!)/A!*B! which result in: ((A+D)(A+D-1)...(A+1))/B!
But now imposing A=(B!)-D the expression is
((B!)(B!-1)...(B!-D+1))/(B!) which is obviously an integer. 

Ex: for C=4
We pick ad es. B=7 and so B-C=3 and A=7!-3=5037
and expression is =5040!/(5037!*7!)=5040*5039*5038/5040=5039*5038.

Edited on February 26, 2016, 5:08 am
  Posted by armando on 2016-02-26 03:18:29

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