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Factorial Ratio (Posted on 2011-02-05) Difficulty: 3 of 5
Prove that the value of [M!/((M+1)(M+2))] is always even, for any given positive integer M, where [x] denotes the greatest integer ≤ x.

** For purposes of the problem, treat zero as an even number.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts just for the record | Comment 1 of 3
<H1 id=firstHeading class=firstHeading>Parity of zero</H1>
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Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. Zero fits the definition of "even number": it is an integer multiple of 2. As a result, zero exhibits the properties shared by all even numbers: 0 is evenly divisible by 2, 0 is surrounded on both sides by odd integers, 0 is the sum of an integer with itself, and 0 objects can be split into two equal groups. Zero fits into the rules for sums and products of even numbers, such as even − even = even, so any alternate definition of "even number" would still need to include zero. Within the even numbers, zero plays a central role: it is the identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively generated. Every integer divides 0, including each power of 2; in this sense, 0 is the "most even" number of all.

  Posted by Ady TZIDON on 2011-02-05 15:29:56
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