Math 126, Number Theory
Final Exam
May 2006
For this take-home final exam, you may use class notes, the text, and other books and
articles. Do not talk to anyone about the test except me. If you have any questions about
the test, ask or email me.
Write the answer to each question on a separate page of paper, then staple the pages
together.
Problem 1. [20] Carefully prove the following statement.
If x=r/s is a rational solution of the equation
anxn+an−1xn−1+· ·· +a1x+a0= 0
where all the coefficients an, an−1, . . . , a1, a0are integers, and if rand sare rel-
atively prime, then rdivides the constant term a0while sdivides the leading
coefficient an.
The proof is not difficult, and it begins by clearing the denominators to turn it into an
equation involving integers only. As you write out the proof, point out each time you use a
property of divisibility or relative primality.
Problem 2. [24; 6 points each part] The Euclidean algorithm has been essential for us this
semester.
a. Choose two of the following numbers, call them mand n, and show how the Euclidean
algorithm is used to find their greatest common divisor d= (m, n).
2907,3128,4807,9775
b. Use the work you did in part a to show how dis a linear combination of mand n, that
is, solve the linear Diophantine equation mx +ny =d.
c. Continuing with this example, show how you can use your results in part b to solve
the linear congruence mx ≡6d(mod n).
d. The Euclidean algorithm also is used to find continued fraction expansions. Find the
continued fraction expansion of m/n, and show your work.
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