Math 2001 Homework #8 Solutions
1. Section 3.2: 11. Suppose a, b, q, and r are nonzero integers such that a = bq + r. Prove
or disprove each of the following statements.
(a) gcd(a, b) = gcd(b, r)
Solution: Importantly, note that we cannot use the Euclidea
Math 2001 Homework #11 Solutions
1. Section 4.5: #7. An integer between 100 and 999 inclusive is selected at random. If
n = 100d2 + 10d1 + d0 has digits d2 , d1 , and d0 , nd the probability that:
(a) The digits of n are all distinct.
Solution: Since 100
Math 2001 Homework #13 Solutions
1. Section 7.1: #3. For each part below, draw a graph with the described properties.
(a) A graph with three vertices and four edges.
Solution: If we have vertices cfw_1, 2, 3, there is no simple graph that has four
Math 2001 Homework #10 Solutions
1. Section 4.1: 6ab. For each map below, determine the number of southerly paths from
point A to point B.
Solution: We just have to use the same process as we did in building Pascals triangle:
mark a 1 to count the paths c
Math 2001 Homework #9 Solutions
1. (a) Roughly how many digits does the number 3893 have?
Solution: Very roughly, we use the fact that 32 = 9 10 and get 3893 310446 ,
so we get something slightly less than 447 digits. More precisely if I plug into
Math 2001 Homework #7 Solutions
1. Section 3.1 #2. Let a, b, c, d be nonzero integers. Prove the following implications.
(a) If a|b and a|c, then a2 |bc.
Solution: Suppose a|b and a|c. Then there are integers j and k such that b = aj
and c = ak. Therefore
Math 2001 Homework #6 Solutions
1. Section 2.1 #11. Suppose m and n are integers. Prove that the following statements
(a) m2 n2 is even.
(b) m n is even.
(c) m2 + n2 is even.
Solution: We need three cyclic proofs: we can either do (a)(b),
Math 2001 Homework #5 Solutions
1. Section 1.5 #4. Determine whether each statement below is true or false. Give the
negation of each statement.
(a) P(cfw_1, 2, 3, . . . , n) P(N) n N
Solution: An element of P(cfw_1, 2, 3, . . . , n) is a subset of cfw_1,
Math 2001 Homework #4 Solutions
1. Section 1.4 #3. Consider the statements R, W , and M below.
R = It rains.
W = Marco goes for a walk.
M = Marco goes to a movie.
Write each of the following statements symbolically. Negate each statement symbolically usin
Math 2001 Homework #3 Solutions
1. Section 1.2 #6: If R and S are subsets of the universal set U , show that R\S = R S c .
Solution: By denition,
R\S = cfw_x R | x S.
The condition that some x U is in R\S is thus that x R and also x S, so another
Math 2001 Homework #1 Solutions
1. (a) Suppose there is a party with 5 guests, and each guest is expected to shake hands
with every other person. How many handshakes happen?
Solution: At rst sight it may seem that each of the ve guests shakes hands