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Unformatted text preview: Monday 04 February 2002
3:00 p.m. — 4:15 p.m. NAME: Id. N0.: Faculty of Mathematics
University of Waterloo MATH 135 Term Test #1
Winter 2002 Calculators are permitted. 00 Please Print l. (a)  Let a and b be integers. Deﬁne what it means for a to divide b. (b)  Determine whether or not each of the following statements is TRUE or FALSE. Write your answer
in the space provided. i.6[3
vi.0[0 2.  Determine whether or not the following statements are equivalent: (P AND Q) => (Q 4:» R) and ((NOT P) 0R(NOTQ))OR(Q e R) 3. Consider the statement Vx3y(:c2 + y < 0). Assume that the domain (universe of discourse) for a: and y is the
real numbers. (a)  Write this statement as an English sentence.
(b)  Is this statement TRUE or FALSE? Justify your answer. (c)  If the domain is changed to the set of positive integers, is the statement TRUE or FALSE? Justify
your answer. 4. (a)  Using the Euclidean Algorithm, calculate GCD(998, 779).
(b)  Find integers a: and y so that 9882? + 7793/ = GC'D(988, 779). 5. (a)  Let a, b and d be integers. State the formal deﬁnition of what it means for d = GC’D(a, b).
b)  Using this deﬁnition, prove that ifa > 0, then GCD(a, 0) = a.
(c)  Suppose a, b, q and r are integers that satisfy b = aq + r. Prove that GCD(a, b) = GCD((1, r). (
(d)  Let a and b be integers with a < b and so that a does not divide b. Consider the following list of
applications of the Division Algorithm: b = (1111 + 1'1 0 S 1'1 < (1
Cl = q2r1+r2 OST2<T1
T1 — (131°2 + T3 0 S 7'3 < r2
Til—2 : tburn—1 + 7'7; 0 S r" < rn_1
rn-l : qn+1rn + 0
Explain why this algorithm (called the Euclidean Algorithm) always terminates and prove that
1",, = GC'D(a, b). 6.  Prove that
13+33+...+(2n—1)3=n2(2n2—1) for all positive integers n.
(Note: Clearly explain all of your steps. Marks for this question will be heavily weighted on your form and
style of presentation.) ...
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