Download Fall 2016 Midterm

MIDTERM ANSWERS
FALL 2016
Individual Part
Question 2 The largest number of steps required by the Euclidean Algorithm in
computing the gcd of two positive integers A and B where each has at most three
digits, is
• 25
• 13
• 18
• 9
• None of the above
Proof. The first few Fibonacci Numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987. The remaining Fibonacci numbers have more than three digits.
By the argument of Lamé’s theorem, the worst case scenario in terms of numbers of
steps for three digit numbers occur with the last two three digit Fibonacci numbers.
Question 3 Which of the following Diophantine Equations is solvable?
• 121 ∗ x + 77 ∗ y = 28
• 29 ∗ x + 123 ∗ y = 5
• 81 ∗ x + 936 ∗ y = 28
• None of the above
Proof. gcd(121, 77) = 11 and 11 6 | 28.
gcd(29, 123) = 1 and 1|5.
gcd(81, 936) = 9 and 9 6 | 28.
Question 4 There are positive integers x and y such that for two distinct numbers
d and D that are not perfect squares, x2 − dy 2 = 1 and x2 − Dy 2 = 1.
• TRUE
• FALSE
Proof. For if for some x and y we have
x2 − d ∗ y 2 = 1
and
x2 − D ∗ y 2 = 1
then subtracting the second from the first gives
(D − d) ∗ y 2 = 0.
Then we either have d = D (thus d and D ar not distinct), or else we have y = 0,
not a positive integer.
Question 5 If φ(k) = 2m for some integer m > 2, then k must be a power of 2.
1
2
FALL 2016
• TRUE
• FALSE
Proof. φ(20) = φ(22 ) ∗ φ(5) = 2 ∗ 4 = 23 , and 20 is not a power of 2.
Group Part
Question 2 Consider the group (U27 , ∗ mod 27). This group has a single element
b such that each element of the group is a power of b mod 21.
• TRUE
• FALSE
Question 3 Which of the following equations requires the largest number of steps in
the continued fractions method to find the solution (x0,y0) with x minimal positive
and y positive?
• x2 − 29y 2 = 1
• x2 − 55y 2 = 1
• x2 − 22y 2 = 1
• x2 − 31y 2 = 1
Proof. Applying the continued fractions algorithm provides the following data:
d
x
y
Steps
29 9801 1820
9
55
89
12
3
22 197
42
5
31 1520 273
7
Question 4 For which of the following pairs (a,b) of positive integers is the number
of steps in applying the Euclidean Algorithm to compute gcd(a,b) the largest?
• (107, 1971)
• (89, 144)
• (15374, 353603)
Question 5 Which of the following groups has the most elements?
• (U29 , ∗ mod 29)
• (S5 , ◦)
• (U113 , ∗ mod 113)
Question 6 Which one of the following has the largest number of distinct prime
factors?
• 256
• 243
• 251
• 217
• 204
• More than one of these numbers has the largest number of distinct pprriimee
factors of them all.
Question 7 Which one, if any, of the following groups is not isomorphic to any of
the others of these groups?
• (Z16 , + mod 16)
• (U40 , ∗ mod 40)
MIDTERM ANSWERS
• (U34 , ∗ mod 34)
• Each of these groups is isomorphic to another of these groups.
3