Assignment 1
Essential Arithmetic
MATH 335 Section 201
Due Wednesday, January 26th at 11 a.m.
NOTE: This assignment is intended to help prepare you for the first midterm. These
problems are based on Vivaldi and Chapter 2.3 in Burger and Starbird.
1. State what a prime number is. Why is there only one even number?
2. Find the prime factorization of 18340489.
3. Use the Euclidean algorithm to find the greatest common divisor of (a) 594 and
693, and (b) 1950 and 7719075. Remember that the Euclidean algorithm makes
use of the fact that for every pair of natural numbers n and m, with m > n say,
we can write m = qn + r, where q and r are natural numbers (with r possibly
0).
4. Find the least common multiple of (a) 48 and 50, and (b) 198 and 242. It will
be useful for you to remember that
lcm(m, n) =
m·n
gcd(m, n)
for m and n natural numbers.
5. Use the sieve algorithm describe by Vivaldi in Chapter I.1 to find all the prime
numbers greater than 74 but less than 169.
6. Goldbach’s Conjecture is a famous unsolved problem that asks: is it true that
every even natural number greater than 2 can be expressed as the sum of two
prime numbers?
(a) Express the first 15 even numbers greater than 2 as the sum of two prime
numbers.
(b) Do you think that every odd number greater than 3 can be written as the
sum of two primes? Is so, try to make an argument to show this is true.
If not, find a small counterexample and show that the given number is
definitely not the sum of two primes.
7. Suppose that p is a prime number greater than or equal to 3. Show that p + 1
cannot be a prime number.