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This homework may be done with a partner. If you work with a partner, submit one copy with both names, along with a description of how you collaborated.

1. Problem 1.12 from the Paar textbook. Show your work.
2. Problem 1.13 from the Paar textbook. Show your work.
3. Problem 1.14 from the Paar textbook. Show your work.
4. Let a, b be integers, m, n be natural numbers with n dividing m. Come up with two examples to show that if a ≡ b (mod m), then a ≡ b (mod n).
5. Find all incongruent solutions modulo the given modulus in each of the following. (That is, your solution will be in the form of x ≡ some number (mod given modulus).)
   1. 3x ≡ 1 (mod 7)
   2. 11x ≡ 5 (mod 13)
   3. -x ≡ 7 (mod 25)

6. Compute 3⁶¹(mod 101) without using a calculator. Show your work.

Submit your homework on Kit.

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