The University of British Columbia
MATH 335 (201)
Midterm 1
25 January 2010
Time: 50 minutes
FULL NAME:
STUDENT # :
SIGNATURE:
This Examination paper consists of 7 pages (including this one). Make sure you have all 7.
instructions:
No memory aids allowed. Only one calculator allowed. No communication devices allowed.
marking:
Q1
Q2
Q3
Q4
Q5
TOTAL
/10
/10
/10
/10
/10
/50
Name of Instructor: Mac Lean
MATH 335 (201) Midterm 1 — 25 January 2010 — p. 2 of 7
Q1
[10 marks]
Short answers: give one or two sentences as responses to the following. The use of examples is
encouraged.
(a) What is the successor in the context of natural numbers?
(b) What is a prime number?
(c) What is the greatest common divisor (gcd) of two numbers?
(d) What is the least common multiple (lcm) of two numbers?
(e) What is the division algorithm?
MATH 335 (201) Midterm 1 — 25 January 2010 — p. 3 of 7
Q2
[10 marks]
(a) Use the Euclidean algorithm to find the greatest common divisor of 900 and 240.
(b) Find the least common multiple of 900 and 240.
(c) Use the Euclidean algorithm to find the greatest common divisor of 264 and 2600.
MATH 335 (201) Midterm 1 — 25 January 2010 — p. 4 of 7
Q3
[10 marks]
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.
MATH 335 (201) Midterm 1 — 25 January 2010 — p. 5 of 7
(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.
MATH 335 (201) Midterm 1 — 25 January 2010 — p. 6 of 7
Q4
[10 marks]
Suppose that p is a prime number greater than or equal to 3. Make an argument to show that
p + 1 cannot be a prime number. (You might find it useful to write down a few examples.)
MATH 335 (201) Midterm 1 — 25 January 2010 — p. 7 of 7
Q5
[10 marks]
Explain the ”Monty Hall” Problem. Give a succinct statement of the problem and then a clear
discussion of the solution.