6. Consider the subring S = {[0], [2], [4], [6], [8], [10], [12], [14], [16] } C Z18 with respect to addition and multiplication defined in Z18, do the following questions. (a) Does S have the unity? (b) Does S have zero divisors? Which element of S have multiplicative inverses?

Q: Prove the following: (a) If a, b,c € Z+ and a+b=c then (c, [a, b]) = (a, b).

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 15. -6,-3, O, 3 find the sum of the frot 15 terms. Show work A15

A: It is given that,the series       -6,-3,0,3,......... We have to find the sum of the first 15 terms.…

Q: 4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b)…

A:

Q: 4. Solve each of these congruences using the modular inverses found in the previous problem. (a) 7x…

A: The given congruence equations are (a)    7x=2 (mod 26) (b)   19x=4  (mod 141)

Q: . Let a, b € Z so that 4a + 5b is a multiple of 13. Prove that 13 divides 9a + 8b.

A: We have to show that 13 divides 9a+8b To prove this we using 4a+5b divisible by 13

Q: why - ( f ²^(lu² (+)² + 9²³\u{(t) ] ²³ ) d+){ 0 is inner Product?

A:

Q: -1 -2 -3 -41 1 2 3 Let A = 1 2.) Find adj (A)

A:

Q: 2. Consider the set of nonzero complex numbers, defined by C* = {a+b√-1: a, b = R} - {0}. For…

A: Show that the inverse of  in  is equal to .The multiplication operation on  defined byWe may assume…

Q: -8 -6 - 41 | + 3. 10 -6] Compute: 4 · 1

A:

Q: Show that [o 2] A 3 0 has no A = L-U factorization where L is lower triangular and U is upper…

A:

Q: Let 71 = 3+7i and 72 = 3– 7i and find z172.

A:

Q: Prove that A2=I.

A:

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

A: The multiplication cx where x∈ℝ2 is defined as cx=cx1, 0 because x=x1, x2 where x1 and x2 belong to…

Q: 2. Transform this LP into its DUAL counterpart. min 3x, + 2x2 - X3 subject to X1 + x2 = 5 -1 < x, <…

A: Transform this LP into its DUAL counterpart. Our problem can also be viewed as,   MIN  3x1 + 2x2 -…

Q: (c) Suppose n EN is divisible by 5. What is (25n) (n) L?

A: Let, the prime factorization of n be as below.          n = p1a1 p2a2 . . . . . . pkak     for some…

Q: 2. Let me N and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integers x…

A: As asked we shall solve only question 2. (a).Let .(a) If gcd(a, m) = 1, then Bézout's lemma gives…

Q: In which of the following sets of numbers does each element have its inverse with respect to the…

A: Consider the given sets with usual multiplication.(a) , (b) , (c) , (d) , (e) .We need to identify…

Q: Find the integer (a+b) with respect to the module 2,3,5 if a=11 and b=13

A:

Q: 3. Let a be a primitive 10th root of unity. Which of the following is false? A. a5 = 1 B. a³ = a¹3…

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Q: Find the product of x and y if x, 3/2, 6/7, y  are in G.P

A:

Q: Suppose that |a| = 24. What is the generator of (a²l)n (a!o)? a20 a a14

A: Suppose that |a| = 24. We have to find the generator of &lt;a21&gt;∩&lt;a10&gt; .

Q: 2.3(i) Find the conjugates, moduli and arguments (to the nearest degree) of the following

A: 2.3(i)   (a). Given,                         3+5i.     To find the conjugate, modulus and argument…

Q: Let Find formulas for the entries of M", where n is a positive integer. Hint: a formula such as 5…

A: The given problem is to find the entries of Mn for the given following matrix M. We have to find the…

Q: Each of the following is incorrect. Explain why or give a counterexample. 1. There are 60 zero…

A: Use the formula of zero divisors in Zm and the properties of Euler's Phi function as in next step.

Q: 5) Find NV ir ZN -- 20 7) Find DW if YW-4

A: Consider that the centroid divides the median in the ratio 2:1 (5) ZNNV=2120NV=21NV=202NV=10

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

A:

Q: a) 2+ 1_3 x´ -9 x+3

A:

Q: 1. Write (2 + i) (4 - 3- 21 in the form a bi, where a,bR.

A: Value of i 2 = - 1 And i 3 = i   ( a - b ) 3 = a3 - b3 - 3 ab ( a - b )

Q: 6. Show that there is a one-to-one correspondence between the Cartesian products A x B x C and A x…

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Q: Write the following expressions in a more compact way (i.e., using index i, j as above). 1. C11 C2x2…

A:

Q: 2. Find the LU factorizations of and 20 5 2 -6 3 -13 -3 4 9 16 17 2 4 -6 2 13 9 دن و ان 5 4

A:

Q: Prove the following: (a) If a, b, c e Z+ and a +b= c then (c, [a, b)) = (a,b). %3D

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 10. Use i = (0, 1) and y = (y, 0) to verify that -(iy) = (-i)y. Thus show that the addi- tive…

A: i=0 ,1y=y ,0 verify :   -iy = -iy  To find additive inverse of z=x+iy