Material for Quiz Three:

Q3.1: 3.1 25-44
Q3.2: 3.2 17, 24, 25
Q3.3: 3.3 24, 25, 30, 36, 41
Q3.4: 3.4 24, 26, 32-35
Q3.5: 3.5 14-24
Q3.6: 3.6 8, 11, 17-23
Q3.8: 3.8 12-14

Hints for homework problems:
3.3 #41.  Think of writing a number in terms of powers of ten, i.e. 235 =
2x100 + 3x10 + 5x1.  Then look at 10^n -1.

Quiz Three:
Average Score: 8.45

1.  Prove the sum of any four consecutive integers is even.
n + (n+1) + (n+2) + (n+3) = 4n+6 = 2(2n+3).  By closure, if n is an integer
then 2n+3 is an integer.  By definition of even, if 2n+3 is an integer
then 2(2n+3) is even.

2.  Prove that for any integer n, n^2 +2 is not divisible by four.
By parity, n is either odd or even.  If n is even, n=2k for some integer
k.  By substitution, n^2 +2 = 4k^2 +2.  4k^2 +2 == 2 mod 4, so 4k^2 +2
is never divisible by four.  If n is odd, n=2K +1 for some integer k.
By substitution, n^2 +2 = 4k^2 +4k +3, which is 3 mod four, so 
4k^2 +4k +3 is never divisible by four.  Finally, by the dilemma principle,
n^2 +2 is never divisible by four.

3.  Find the unique prime decompositions for 252 and 2970.  Use them to
find the GCD of 252 and 2970.  Then use the Euclidean algorithm to check
your answer.
252 = 2^2 x 3^2 x 7; 2970 = 2 x 3^3 x 5 x 11  The GCD is the product of
all the powers they have in common; in this case 2 x 3^2 = 18.

2970 = 11 x 252 + 198
252  = 1  x 198 +  54
198  = 3  x  54 +  36
54   = 1  x  36 +  18
36   = 2  x  18 +   0