Math 115A Homework Assignment #1, Due 10/3
1
Required Problems
From Section 1.1 in the text:
1. (b) Show that 1 + 3 + 5 + + (2n 1) = n2 for all positive integers n.
1. (e) Prove Nichomachus theorem by induction, i.e., 13 + 23 + 33 + + n3 =
n(n+1)
2
2
for
MATH 115A SOLUTION SET V
FEBRUARY 17, 2005
(1) Find the orders of the integers 2, 3 and 5:
(a) modulo 17;
(b) modulo 19;
(c) modulo 23.
Solution:
(a) Eulers theorem implies that it suces to consider exponents which are divisors of 16.
Working modulo 17 gi
MATH 115A PROBLEM SET VI
FEBRUARY 17, 2005
(1) Find the index of 5 relative to each of the primitive roots of 13. [Hint: Recall that
2 is a primitive root modulo 13. To nd the other prinitive roots, use the table that was
written down in class.]
(2) Find
MATH 115A SOLUTION SET II
JANUARY 20, 2005
(1) Use the Euclidean algorithm to nd the highest common factor of 18564 and 30030.
Check your answer by writing each number as the product of prime powers.
Solution:
Applying the Euclidean algorithm yields the f
MATH 115A SOLUTION SET IV
FEBRUARY 10, 2005
(1) Suppose that f and g are multiplicative functions. Prove that the function F dened
by
F (n) =
f (d)g(n/d)
d|n
is also multiplicative.
Solution:
Let m and n be relatively prime positive integers. Then
F (mn)
MATH 115A PROBLEM SET II
JANUARY 13, 2005
(1) Use the Euclidean algorithm to nd the highest common factor of 18564 and 30030.
Check your answer by writing each number as the product of prime powers.
(2) Which of the following Diophantine equations cannot
MATH 115A SOLUTION SET VI
FEBRUARY 24, 2005
(1) Find the index of 5 relative to each of the primitive roots of 13. [Hint: Recall that
2 is a primitive root modulo 13. To nd the other prinitive roots, use the table that was
written down today in class.]
So
MATH 115A SOLUTION SET III JANUARY 27, 2005 (1) Use Fermat's Little Theorem to verify that 17 divides 11104 + 1. Solution: Working modulo 17, we have that 11104 (112 )52 252
5 10
(mod 17) (mod 17) (mod 17)
(mod 17)
10
(2 ) 22 (-2) 4 44
(mod 17)
16 (mod
MATH 115A PROBLEM SET V
FEBRUARY 10, 2005
(1) Find the orders of the integers 2, 3 and 5:
(a) modulo 17;
(b) modulo 19;
(c) modulo 23.
(2) Establish each of the following statements below:
(a) If a has order hk modulo n, then ah has order k modulo n.
(b)
MATH 115A PROBLEM SET I
JANUARY 6, 2005
1 (i) Suppose that n > 1 is a composite integer, with n = rs, say. Show that 2n 1 is
divisible by 2r 1. (This shows that 2n 1 is prime only if n is prime. Primes of the form
2n 1 are called Mersenne primes.)
(ii) Sh
MATH 115A PROBLEM SET IV
JANUARY 27, 2005
(1) Suppose that f and g are multiplicative functions. Prove that the function F dened
by
F (n) =
f (d)g(n/d)
d|n
is also multiplicative.
(2) (i) For each positive integer n, show that
(n)(n + 1)(n + 2)(n + 3) = 0
MATH 115A PROBLEM SET III
JANUARY 20, 2005
(1) Use Fermats Little Theorem to verify that 17 divides 11104 + 1.
(2) Show that for any integer n 0, 13 | (1112n+6 + 1).
(3) Let a be any integer. Show that a and a5 have the same last digit.
(4) Use Fermats Li