MAT246 SPRING 2009 L0101
PROBLEM SET 1 SOLUTIONS
(1) (p. 17, #2) Prove by induction:
n
1
3
n
n+2
2
i
= + 2 + 3 + + n = 2 n .
i
2
22
2
2
2
i=1
Solution: In the base case, n = 1, the statement reduces to
1
3
=2 ,
2
2
which is true. Now assume the statement
MAT246 SPRING 2009 L0101
PROBLEM SET 3 SOLUTIONS
(1) What is the remainder when 11 18 2322 13 19 is divided by 7?
Solution: 11 18 2322 13 19 4 4 5 6 5 (4 5)2 6 63 6.
(2) What is the remainder when 1 + 2 + 22 + + 2219 is divided by 5? (There are a couple o
MAT246 SPRING 2009 L0101
PROBLEM SET 7
(1) The number 1 has six sixth roots in the complex plane. What are they, in the form a + bi?
Solution: In polar coordinates, 1 = (1, 2k ) where k is any integer. Thus the sixth roots of
1 (in polar form) are precise
MAT246 SPRING 2009 L0101
PROBLEM SET 9 SOLUTIONS
(1) We formally dened (will dene?) the set of constructible objects C in Fridays lecture. Prove that
this set is countable. (Hint: For the hard direction, note that the set of objects dened at any stage
is
MAT246 SPRING 2009 L0101
PROBLEM SET 11
(1) Prove that x6 x2 + 2 has no constructible roots.
Solution: Suppose a is a constructible number where a6 a2 + 2 = 0. Let b = a2 ; then b is
constructible and b3 b + 2 = 0. But f (x) = x3 x + 2 has no rational roo
MAT246 SPRING 2009 L0101
PROBLEM SET 10 SOLUTIONS
(1) Let F be a subeld of R. Consider the system
(x a)2 + (y b)2 = c
(x d)2 + (y e)2 = f
(1)
where a, . . . , f F . If (x, y ) is a solution to this system, show that x, y E , where either E = F
or E = F k
MAT246 SPRING 2009 L0101
PROBLEM SET 8 SOLUTIONS
(1) Prove that ( ) = for all cardinalities , , and . Also prove that, if , then .
(First interpret the statements in terms of sets A, B , and C ; the necessary functions should then be
clear.)
Solution: By
MAT246 SPRING 2009 L0101
PROBLEM SET 5 SOLUTIONS
(1) Suppose you are eavesdropping on Alice and Bob. Bob sends Alice the encrypted message 1130. If
Alices public key is (N, e) = (1643, 223), what is the intended message? (You may want to do the
next probl
MAT246 SPRING 2009 L0101
PROBLEM SET 6 SOLUTIONS
(1) Express
3
1
+i
2
2
3
,
1
,
(2 + i)(1 3i)
and
5 + 12i
in the form a + bi, with a, b real. (See the text, page 91.) Also express (3 +
form a + b 2, with a, b rational.
2)/(1 +
2) in the
Solution:
1
3
+i
MAT246 SPRING 2009 L0101
PROBLEM SET 4 SOLUTIONS
(1) For positive natural numbers a, b, let lcm(a, b) be the smallest natural number c where a | c and
b | c. (Certainly such numbers exist; see ab.) Show that
ab
lcm(a, b) =
.
(a, b)
Solution: Let cfw_p1 ,
MAT246 SPRING 2009 L0101
PROBLEM SET 2
(1) We dene the triple primes as triples of natural numbers (n, n + 2, n + 4) for which all three entries
are prime. How many triple primes are there? (Hint: mod 3.) (By way of contrast, it is not yet
known whether t
MAT246 SPRING 2009 L0101
PRACTICE FINAL EXAM (FROM FALL 2008)
There are 10 problems, worth 10 points each.
(1) (a) Calculate (20100 ), where is the Euler function.
(b) Find an integer x such that 140x 133 (mod 301). Hint: gcd(140, 301) = 7.
+1)
for every
MAT246 SPRING 2009 L0101
PRACTICE FINAL EXAM (FALL 2008) SOLUTIONS
(1) (a) Calculate (20100 ), where is the Euler function.
(b) Find an integer x such that 140x 133 (mod 301). Hint: gcd(140, 301) = 7.
Solution:
(a) (20100 ) = (2200 5100 ) = (2200 ) (5100
MAT246 SPRING 2009 L0101
FINAL EXAM SOLUTIONS
(1) Prove that for all natural numbers n,
12 + 32 + 52 + + (2n + 1)2 =
(n + 1)(2n + 1)(2n + 3)
.
3
Solution: For the base case n = 0 (n = 1 is also acceptable), we must check that
113
12 =
;
3
both sides are e
MAT246 SPRING 2009 L0101
FALL 2008 MIDTERM SOLUTIONS
(1) Prove, by mathematical induction, that n3 + 5n is divisible by 6.
Solution: I will rst assume the question means for all n 0. We have a base case n = 0, whence
n3 + 5n = 0 is divisible by 6. Assume