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Prime Test (2) (Posted on 2011-05-08)

Let S be the cube of a prime number, such that S is greater than 504.

Prove that (S-125)(S+125) is evenly divisible by 504.

See The Solution Submitted by broll

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Solution

Comment 3 of 3 |

Let S = p³ where p is prime and p >= 8.

Let   y = (S - 125)(S + 125)

            = p⁶ - 15625

            = p⁶ - 7*8*9*31 - 1

Since p is not a multiple of 7:      p⁶ = 1 (mod 7), so y = 0 (mod 7)

Since p is odd:                           p⁶ = 1 (mod 8), so y = 0 (mod 8)

Since p is not a multiple of 3:      p⁶ = 1 (mod 9), so y = 0 (mod 9)

Since 7, 8 and 9 have no common prime factors, it follows that y is
divisible by 7*8*9 (= 504).

Posted by Harry on 2011-05-13 15:04:09

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