1(d) -162=(17)(-10) + 8, so q=-10 and r=8. Remember that 
.
2(b) 39214=(871)(45)+19 so r=19=39214-(871)(45).
6. S'pose the three consecutive integers are a, a+1 and a+2. If a is divisible by 3, then we're done. Assume 3 does not divide a. Then use the division algorithm to write
with r=1 or r=2. If r=1 then
(add 2 to both sides of the other equation). Since 3 divides the right hand side, 3 divides a+2. Similarly, if r=2 then 3 divides a+1. Therefore, 3 must divide one of a, a+1 or a+2.
9(b). Use the Euclidean Algorithm:
Therefore, gcd(41, 25)=1 (you could also do this using the prime factorization of 41 and 25). Working backwards, we have
44. We show the representation exists by strong induction on n. If n=0, take 
. If n>0, write 
with 
using the division algorithm. then
and 
because q<0 implies 
, contrary to the hypotheses. Hence by induction, 
, with 
for all i. The representation of n now follows because . To prove uniqueness, suppose
is another such representation Then 
and 
are each the remainder when n is divided by b, so 
. Now 
, so 
follows the same way. The continues to show 
for all 
.