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A Composite Determination Problem (Posted on 2006-04-23) Difficulty: 3 of 5
Determine whether or not N is a composite number, where

N = 675*2621 + 677*2610 - 1

A prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The numbers 0 and 1 are considered to be neither prime nor composite.

See The Solution Submitted by K Sengupta    
Rating: 4.3333 (3 votes)

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Solution Another Stab at the Solution | Comment 3 of 14 |
OK, how about this:

675 is a multiple of 3.  Multiplying by 26^21 doesn't change this fact.

677 is one less than a multiple of 3.  Each time you multiply by 26, the product alternates between being one more and one less than a multiple of three.  For example:

677 = 3n-1
677*26^1 = 3n + 1
677*26^2 = 3n - 1
677*26^3 = 3n + 1

So, 677*26^10 will be one more than a multiple of three.  This one is then subtracted out, so the value of N must be divisible by three and therefore composite.

  Posted by tomarken on 2006-04-23 11:05:47
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