Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute [5]-1 o [12] in Z17.

Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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Q: Let a, m € Z+ and (a,m) = (a - 1, m) = 1. Then prove that 1+a+...+ a(m)-1 = 0(modm)

A: For given,       (a,m) = (a -1, m) = 1 We have to show,      1 + a + a 2 + ........ + a φ (m)-1 ≡ 0…

Q: Prove that 14n is not O(12n)

A: Note that a function f(n) is big O(g(n)) if f(n) is less than constant times g(n) for large n. This…

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: . Determine whether or not x³ + 2x + 1 and x4 + 3x - 2 are congruent mod (x² + 2x + 2) in Q[x].

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Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: Find all Pnterers x such that 5x=12 (mat) LoxE 4 (mad) 7x=16 (modlo) and 1o00s x3000

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Q: 5. If ged(a, n) = 1 then the order of a in (Z/nZ)× is the smallest integer i such that a¹ = 1( mod…

A: Since gcd(a, n) = 1, then             aφ(n) ≡ 1 (mod n)  Now, let order(a) = k.        => by…

Q: a) Find integers a and b such that a + b =a-b (mod 5).

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Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

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Q: Let n be a number with 3 digits. (i.e 100sns 999) Show that if the sum of digits of n is divisible…

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Q: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

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Q: Determine a positive integer x such that 57x +7 = 78 (mod53).

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Q: Is 3 a primitive root of 43? Use your answer to determine all positive integers less than 43 whose…

A: To determine whether 3 is a primitive root of 43, we need to check if it generates all the positive…

Q: (b) Compute 9-¹ (mod 19) by listing the multiples. Show your work.

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Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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Q: For which positive integers a is the congru

A: Given,               ax4≡2 (mod 13)

Q: x² + x = 13(mod17) X= * (mod17)

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Q: Are the polynomials x³+2x +2 and x4 + 3x -2 equal to mod(x² + 2x + 2) in Q[x]?

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Q: Show that y = x (mod p) leads to -y = -x (mod p).

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Q: 6. Let m be an odd positive integer. Prove that every integer is congru- ent modulo m to one of the…

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Q: Let

A: We construct multiplication table for Z3[i]

Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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Q: Use the PMI to prove that 6" – 1 is divisible by 5 for every integer n > 0.

A: To prove that 6^n-1 is divisible by 5 for every integer greater that or equals to 0.

Q: Q7 Compute 59^11 mod 13 Note: You should be able to do this using the modulo squaring method and…

A: 59^11 mod13

Q: Prove that for every odd prime 1000 < p < 2000 the number of –2, –1,2 which are squares mod p is not…

A: To prove that for every odd prime numbers strictly between 1000 and 2000 the numbers -2,-1, 2 which…

Q: Assume that an+1 √12+ an with ao = √12. Use induction to show tha an < 4

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Q: Let d = (4087, 4757). Writed as a linear combination of 4087 and 4757, and find the multiplicative…

A: Apply division algorithm, to find the gcd of 4087 and 4757.

Q: Let A = {1, 2, ..., 12}. Let aRb mean a = b (mod 5). (a) Give [2]R. [3]R, and [5]R. (b) Give [2]…

A: A set of ordered pairs is referred to as a relation in the area of mathematics and computer science.…

Q: Prove the following for Integers x and y, (17 | 10x + 8y) = (17 | 3x + 16y)

A: 17 | 10x + 8y where x and y are integers =&gt; 10x + 8y = 17m for some integer m

Q: 5. Let a and b be positive integers such that gcd(a, b+1) = 1 and b = 1 (mod a).What is the…

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