1.2 Divisors and Prime Factorization
1.
a. 391=23·17+0
c. −116=(−9)·13+1
2.
a. n/d=51837/386=134.293, so q=134. Thus r=n−qd=113.
3. If d>0, then 
d=d and this is the division algorithm. If d<0, then d=−d>0 so n=q(−d)+r=(−q)d+r, 0≤r≤d.
d=d and this is the division algorithm. If d<0, then d=−d>0 so n=q(−d)+r=(−q)d+r, 0≤r≤d.
5. Write m=2k+1, n=2j+1. Then m2−n2=4[k(k+1)−j(j+1)]. But each of k(k+1) and j(j+1) is even, so 8 
(m2−n2).
(m2−n2).
7.
a. 10(11k+4)−11(10k+3)=7, so d 7. Thus d=1 or d=7.
7. Thus d=1 or d=7.
9.
11. If m=qd, then , so . Similarly, . If d=xm+yn, then , so any common divisor of and
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