Suppose that 3x ≡ 2 (mod 5). Find the remainder when x is divided by 5; in other words, compute x (mod 5)

Use your work from problem 2 and the method used in the example at the end of Lecture 16 to solve the congruence: 7x5 is congruent to 6 (mod 17) .

Show that every nonzero residue (mod 17) can be written as a power of 3.

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List all the possible orders of a number x (mod 53) and calculate this order when 10.

Find all solutions for 10x = 14 mod 1008

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8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10. Apply Fermat's Little Theorem to find the remainder when 22⁰ + 330 + 440+550 + 660 is divided by 7.

solve this congruence with the help of modular inverses: 89x = 2 (mod 232)

Solve the congruence8x= 16 (mod 424).  If none exists, explain why.  If there are one or moresolutions, give a complete list of solution(s)