1. Simplify the following expressions. Your answers should be in the range .
2. If , the multiplicative inverse of x mod n is denoted . It is the number (if there is one) which satisfies the equations
(a) Find . Your answer should be in the range .
(b) Show that 4 does not have a multiplicative inverse mod 6.
(a) Since , it follows that .
(b) One approach is to try all the numbers in :
Alternatively, you can give a proof by contradiction. Suppose . Multiply both sides by 3:
The last line is a contradiction, and so there is no number x such that .
3. Use modular arithmetic to prove that if , then is not divisible by 5.
If , then , 1, 2, 3, or 4 mod 5. I take cases:
For all x I have . It follows that if , then is not divisible by 5.
4. Give specific values of and with to show that the following statement is false:
"If , then ."
Take , , , and . Then . However, and , and .
Nothing is so exhausting as indecision, and nothing is so futile. - Bertrand Russell
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