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
(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:
(b) and , but .
5. Let G be a group and let . Suppose that and
Prove that .
Since , I have
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