For each divisor k>1 of n, let U,(n) = {x E U(n) | x mod k = 1}. [For example, U,(21) = {1,4, 10, 13, 16, 19} and U,(21) = {1, 8}.] List the elements of U,(20), U,(20), U,(30), and U1,(30). Prove that U(n) is a subgroup of U(n). Let H = {x€ U(10) |x mod 3 = 1}. Is Ha subgroup of U(10)? (This exercise is referred to in Chapter 8.)

Q: Find the first three values of the linear congruential generator X; = 2x;–1 +3 (mod 13) starting…

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Q: Let i be a positive integer and A, [-i+ 1,1]. Then U14, %3D O R O None of these O F1,11

A: Given that 'i ' be a positive integer and Ai=-i+1,1. We have to find the union ∪i=1∞ Ai.

Q: [1 F 3.11 |8 E 5||1 8 [4 _G 11 [1 4 1] 3 2] [4 2 1]

A: given A=01B=10C=11D=00 E=AT=01 F=BT=10 G=CT=11H=DT=00 K=101L=001M=KT=101N=LT=001

Q: 3. Let m = 15. Compute (d) for every divisor d of m, and check that Edm (d) = m. Repeat this…

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Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n +…

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Q: Consider the set A={ Identify An B. 3n+2 4n-1 4n+1 {In € N*}. 5n-3 In € N*} and B={

A: Kindly refer to Step 2 for the solution.

Q: Compute f2,f3, and f-1 for each of the following permutations.   f= (l,3,7,4)(2,5,9,8,6)

A: f= (l,3,7,4)(2,5,9,8,6)

Q: Let Find formulas for the entries of M", where is a positive integer. M" O M -2*5**n 10 5 -5*n

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Q: Prove that if r <s < n, then P(n , s) is divisible by P(n,r).

A: Given: r≤s≤n Therefore, n-r≥n-s So we can write n-r! = n-rn-r-1n-r-2...n-sn-s-1n-s-2...1 n-r! =…

Q: Let an = an-1+ an-2 with ao = 0, a1 = 1. Show that an is even iff ( if and only if) n is a multiple…

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Q: compute η2

A: η2  is given by : η2=SSbetweenSStotal

Q: k = 4 4. Let k be the second smallest nonzero digit of your Student ID. Compute the order of (k, k)…

A: To find the order of (4,4) .

Q: Consider the set Z, × Z, = {(x, y) : x € Z, and y € Z3}. Define the operation/sum O in Z2 × Z3 such…

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Q: Find all 6 permutations of (12)(345).

A: Find all permutations with 12 in the first pair of bracket : (12)(345) (12)(354) (12)(435) (12)(453)…

Q: Show that a,b,c are in A.P. If x is the G M be tween a and b and y is the G M between b and c,  then…

A: We have to show if  a, b, c are in A.P.,  x is the G M be tween a and b and y is the G M between b…

Q: {1,3, 5, 7, 9, 11}. Use the Let X = {4, 5, 6, 7, 8, 9} and Y roster method to describe each of the…

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Q: nsist only of of even length. Generalize to find the EGF number of ations that consist only of…

A: To find: (1) The EGF for the number of permutations that consist only of cycles of even length.…

Q: If gcd(7, a) = 1, prove that either a +1 or a – 1 is divisible by 7. 20

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Q: rove that if r < s < n, then P(n, s) is divisible by P(n,r).

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Q: Then, there exists k1≠0, k2≠0  such that a,b=k1c,k1d   and  c,d=k2e,k2f. a,b=k1c,d=k1k2e,…

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Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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Q: If a =b (mod m) then a" = b" (modm) Vn e Z*.

A: Given a≡bmod m. we have to show that an≡bnmod m. we wil prove this by result by using method of…

Q: Let x € Z. Prove that if x² - 8x + 11 is divisible by 4, the x6- 1 is also divisible by 4.

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Q: 1. Show that for all z1, Z2 E C, Then we have: |Z1 + z2| < ]z1]+ |z2| 2. Show that for all z1, Z2 E…

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Q: 1. For given fuzzy number A={(3/0,4); (4/0,6)(3/0,8)(4/1)}; B={(2/0,7):(3/1): (4/0,7)}. Determine A+…

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

Q: Find ntegers G and t such that gcd (963,652)3967+t-652,

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Q: 1. List a pair of zero divisors of (M₂(Z), +, ×):

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Q: Let a = (17 3)(5 9 4 2) and B = (2 3)(7 4)(5 1 8) be elements of S10. Then, a'B is O (1 4)( 27 58 9…

A: Alpha square beta inverse is : (1 4)(2 7 5 8 3 9) Third option is correct.