# Solutions to Problem Set 7

Math 345-504

2-9-2018

1. Find the greatest common divisor of 831 and 240.

Then write as a linear combination of 831 and 240 with integer coefficients.

2. (a) Find and write it as a linear combination with integer coefficients of 103 and 83.

(b) Using the result of (a) and without using trial and error, find specific integers x and y such that

(a)

(b) Multiplying the equation in (a) by 55, I obtain

Thus, , is a solution to the equation.

In fact, there are infinitely many solutions of the form:

3. Prove that if n is an integer, then and are relatively prime.

4. (a) Let . Prove that if and and , then .

(b) Give a counterexample to show that if and and , it does not necessarily follow that . (Note the difference in assumptions between (a) and (b)!)

(a) If and , then and for some .

If , there are integers a and b such that

Multiply by x and substitute:

Hence, .

(b) and , but .

MATH 504]

5. Let G be a group and let . Suppose that and

Prove that .

Since , I have

Then

It is now, and in this world, that we must live. - Andr\'e Gide

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