# Solutions to Problem Set 24

Math 310-01/02

11-15-2017

1. (a) Find (that is, find the number n in which satisfies the equation ).

(b) Use the result of (a) to solve the equation . Your answer should be a number in the range .

(a) , so .

(b)

2. Prove that there is no number x such that .

(This shows that 9 does not have a multiplicative inverse mod 99.)

Suppose . Multiplying by 11, I get

This contradiction shows that there is no such x.

3. Define by

Prove that f is bijective by constructing an inverse .

Demonstrate that your inverse works by showing that and .

First, I'll work backwards to "guess" the inverse. If , then . Substitute the definition of in :

The second equation gives . Then

So

Next, I'll check that the guess works:

This proves that is the inverse of f, so f is bijective.

He who has not lost his head over some things has no head to lose. - Jean-Paul Richter

Contact information