I. Solve for x. All final answers must be in Znwhere nis the modulus.  (a) x≡3(15) + 45 mod 18 (b) x≡−12 mod 13 (c) 6xmod 11 ≡1 (d) x+ 2 mod 9 ≡0 II. Write down the congruence classes formed, modulo 4.  III. Find a multiplicative inverse of 3 modulo 6, if there are any.

Q: Let |a|=30.  How many left cosets of <a4>  in <a> are there?  List them.   I was able to…

A: Given that,|a|=30One left coset is <a4>.

Q: to show that the numbers in question are incongruent modulo n.] 11. Verify that 0, 1, 2, 22, 23,…

A: We shall solve this question in next step.

Q: If x98 = 1(mod 97) then possible value of x is of (a) 97 (b) 194 (c) 90 (d) 291 (е) None of these O…

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Q: 7. Find the modulus and argument of -7 + 8i. Hence, express (-7+ 8i)8 in the form x + yi.

A: Answer

Q: Let a, n be positive integers. Knowing that 15a - 13n = 1, we deduce that the multiplicative inverse…

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Q: 3. Solve the following congruences. † (a) 10x = 5 (mod 21) (b) 10x = 5 (mod 15) † (c) 10x = 4 (mod…

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Q: Let Z be the set of all integers and Q be the set of all rational numbers and R be the set of all…

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Q: Evaluate the expressions. (a) 50 div 7 7.143 (b) 50 mod 7 1 ]x

A: We shall answer the following in next step. We also recall the definitions

Q: Prove that if a = b mod n and c= d mod n, then ac bd mod n.

A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…

Q: Number theory

A: