**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 *q _{1}, q_{2}, q_{3}*,
and

x_{1} = q_{1} |

x_{2} = q_{1}+1/q_{2} |

x_{3} = q_{1}+1/(q_{2}+1/q_{3}) |

x_{4} = q_{1}+1/(q_{2}+1/(q_{3}+1/q_{4})) |

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 *x _{i}* are
known as