Three Additional Exercises for Chapter 5...
1. Prove the following theorem:
Theorem: If a, b, m, and n
are integers, and if c | a and c | b, then c | (ma
+ nb).
2. Prove that (a + cb, b) = (a, b) for all integers a, b, and c.
3. Let a = 314159 and b = 100000. Using a calculator or MAPLE, carry out the first four steps of the Euclidean Algorithm for gcd(a, b) to determine the quotients q1, q2, q3, and q4. Then calculate the rational numbers:
| x1 = q1 |
x2 = q1+1/q2 |
| x3 = q1+1/(q2+1/q3) |
| x4 = q1+1/(q2+1/(q3+1/q4)) |
and compare them with the value of a/b, 3.14159, which is the decimal expansion of pi correct to five decimal places.
(Note: Rational numbers of the form xi are known as continued fractions. There has been much written about such fractions.)