== A very short note of HW#5 == 
1, true 
  suppose n is odd then
   1) n is odd;
   2) n * n is odd;
   3) (n * n) * n is odd;
   4) the sum of 3 odd numbers is odd;

OR
   let n = 2k + 1
   plug it into n + n ^ 2 + n ^ 3


2, true

  2 * n ^ 2 - 5 * n + 2 = (2 * n - 1) * (n - 2)
   (2 * n - 1) * (n - 2) is prime
     if and only if 2 * n - 1 = 1 or n - 2 = 1
       of 2 * n - 1 = -1 or n - 2 = -1
   thus, n = 1 or n = 3 or n = 0
   however, for n = 1, 2 * n ^ 2 - 5 * n + 2 = -1
   so n = 3 or n = 0

3, true 
  if a = b (mod q) then there exists an integer k, such that
    a = q * k + b
   p|q implies there is an integer m such that
    q = p * m
   so
    a = p * (m * k) + b
    a = b (mod p)

4, false 
  (-1) ^ 2 = 1

5, true 
  n * (n + 1) * (n + 2) * (n + 3)
   = (n * (n + 3)) * ((n + 1) * (n + 2))
   = (n ^ 2 + 3 * n) * (n ^ 2 + 3 * n + 2)
  let w = n ^ 2 + 3 * n + 1
   (n ^ 2 + 3 * n) * (n ^ 2 + 3 * n + 2)
   = (w - 1) * (w + 1)
   = w ^ 2 - 1
  so
  n * (n + 1) * (n + 2) * (n + 3) 
  = (n ^ 2 + 3 * n + 1) ^ 2 - 1

6, false
  4 | n * (6 * n + 16)
  iff, 4 | n * (2 * n + 0) 
  iff, 4 | 2 * n ^ 2
  iff, 2 | n ^ 2
  but n ^ 2 is odd.

7, true
   a | b iff, b = a * k
   a | c iff, c = a * m
   so
   b - c = a * (k - m)
   a | (b - c)