Determine whether or not N is a composite number, where
N = 675*26^{21} + 677*26^{10}  1
NOTE:
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.
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 = 3n1
677*26^1 = 3n + 1
677*26^2 = 3n  1
677*26^3 = 3n + 1
etc.
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 20060423 11:05:47 