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 .
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
Next, I'll check that the guess works:
This proves that is the inverse of f, so f is bijective.
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