Prof. John Beachy MATH 420, Section 1 8/28/06
Homework 1 due Friday, September 1, 2006
Hand in: pages 14–15, #7, 10, 16, 21
7. Let a, b, c be integers. Give a proof for these facts about divisors:
(a) If b|a, then b|ac.
(b) If b|aand c|b, then c|a.
(c) If c|aand c|b, then c|(ma +nb) for any integers m, n.
10. Let a, b, c be integers, with c6= 0. Show that bc |ac if and only if b|a.
16. Let a, b, c be integers, with b > 0, c > 0, and let qbe the quotient and rthe remainder when ais divided by b.
(a) Show that qis the quotient and rc is the remainder when ac is divided by bc.
(b) Show that if q0is the quotient when qis divided by c, then q0is the quotient when ais divided by bc. (Do
not assume that the remainders are zero.)
21. Prove that the sum of the cubes of any three consecutive positive integers is divisible by 3.
Quiz 1 on Friday, September 1, 2006
You need to know the following definitions and statements of results: 1.1.1 through 1.1.5.
Questions will be taken from the following list of problems: pages 13–14, #2, 8, 9, 15, 17, 18
2. Find the quotient and remainder when ais divided by b.
(a) a= 99 , b = 17
(b) a=−99 , b = 17
(c) a= 17 , b = 99
(d) a=−1017 , b = 99
8. Let a, b, c be integers such that a+b+c= 0. Show that if nis an integer which is a divisor of two of the three
integers, then it is also a divisor of the third.
9. Let a, b, c be integers.
(a) Show that if b|aand b|(a+c), then b|c.
(b) Show that if b|aand b6 | c, then b6 | (a+c).
15. Give a detailed proof of the statement in the text that if aand bare integers, then b|aif and only if aZ⊆bZ.
17. Let a, b, n be integers with n > 1. Suppose that a=nq1+r1with 0 ≤r1< n and b=nq2+r2with 0 ≤r2< n.
Prove that n|(a−b) if and only if r1=r2.
18. Show that any nonempty set of integers that is closed under subtraction must also be closed under addition.