The elements of order 4 modulo 29 are O 6 and 12 O 7, 14 and 21 O 12 and 17 None of the choices

Q: 9. Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

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Q: The number 73 is prime. Enter an a such that 0 <a < 73 and a = 6 (mod 73), if such an a exists.…

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Q: if b is modulo m and a is another unit with a=b mod m, we say that d is the---------of a modulo m to…

A: As with such a large amount of concepts we are going to see, congruence is straightforward, perhaps…

Q: If n is odd integer such that (n, 21) = 1, then n4 = 1 mod 6 n2 = 1 mod 24 This option This option…

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Q: 9. Let a and b be integers. Prove that if a =7 (mod 8) and b = 3 (mod 8), then: (a) a + b = 2 (mod…

A: Your questions solution below in images

Q: 2. Give an example to show that a? = b2 (mod n) need not imply that a =b (mod n). 3. If

A: Given below the detailed solution

Q: an is an even number, show that a2 = 0 (mod 4), and if an is an odd number, show that a2 = 1 (mod…

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

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Q: Prove the statement by contradiction.

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Q: If n is odd integer that is not divisible by 3, then n6 = 1 mod 168 n2 = 1 mod 24 O This option O…

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Q: The elements of order 4 modulo 29 are O 6 and 12 O 7, 14, and 21 O 12 and 17 O None of the choices

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Q: 5- If a = b mod(n), then ac = bc mod(n). 6- If a b mod (n), then ak = b*mod(n) for every positive…

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Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: If a = b (mod m) and a = b (mod n), and gcd (m,n) = 1, then a = b (mod mn) Select one: O True O…

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Q: 5. Let p = 61. Find all incongruent elements modulo 61, that have order: (a) 5 (b) 10 (c) 60

A: p=61 incongruent modulo 61={x:1≤x≤n with gcd(x,n)=1}={1,2,3,4,5,6,7,8,9,10,11,12,13,...60}

Q: 5. Let p = 61. Find all incongruent elements modulo 61, that have order: (а) 5 (b) 10 (с) 60

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Q: The order of the integer 4 modulo 19 is: O 4 O None of the choices O 6

A: To find The order of the integer 4 modulo 19.

Q: 2. In Section 3.1, we defined congruence modulon where n is a natural number. If a and b are…

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Q: If n| m and a = b(mod m), then a = b(mod n).

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Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: The last two digits of 52044 is 44. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). F 10. F…

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Q: For all integers a and b, if a = 7 ( mod 8)and b = 3 ( mod 8), then (a + b) = 2( mod 8).

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Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

Q: Proposition 7. Let a, b, c, d E Z and n be a positive integer. If a = b (mod n) andc = d (mod n),…

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Q: 3) Show that 1-2+2-3+3-4+…+(n – 1) - n is O(n³). Also state the values for witnesses C and

A: Hi! You have posted multiple questions. As per norms, we will be solving only one question. We will…

Q: Which of the following are true modulo 12? (1) Does 5z = 6 mod 12 have a solution? How many? (2)…

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Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

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Q: The order of 2 mod 17 is: a) 2 b) 4 c) 8

A: The order of a modulo p then is the least positive integer n such that an ≡ 1(mod p) where p is a…

Q: (c) For all integers a and b, if a · b = 1 (mod 6), then a =1 (mod 6) or b = 1 (mod 6). (d) For all…

A: (c) The statement is false. For example take a=5, b=5 ab=2525≡1 mod(6) Hence ab≡1 mod(6) But a≡5…

Q: Ha = b (mod n) and gcd (a,n)= d, then gcd (b,n) %3D

A: Given a≡b(mod n). This can be written as a-b≡0(mod n). Implies a-b=kn for some natural number k.…

Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

Q: If a = b (mod n) and b = c (mod n) then which of the following is false. a =c (mod n) a = c (mod n)…

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Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

Q: 4.14. Let a, b, n E Z, where n > 2. Prove that if a = b (mod n), then a2 = b (mod n). 4.15. Let a,…

A: The detailed solution is as follows below:

Q: If a has order 2k, then ak =? mod p a) 0 b) 1 c) -1 O b a

A: Solution

Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

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Q: (f) If a = b (mod n), then a* bk (mod n) for any positive integer k.

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Q: If a = b (mod 5), then... Choose all correct statements. O a3 = b³ (mod 5) O a + b = 0 (mod 10) O a…

A: Option (a), (c), and (d) are correct options.

Q: The order of the element 16+ in Z24/ is O 8 O 4 O 1 O None of the choices O 2

A: Let (G,*) be the group and let N be the normal subgroup of G then the set of all left or right…

Q: 20. If a e Z and a = 1 (mod 5), then a? = 1 (mod 5).

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Q: Q2/let n be a fixed positive integer and a,b be arbitrary integers. Then a = b (modn) iffa and b…

A: We have to prove that Given statement

Q: If 3(26!) = x(mod29), then x = None of the choices O 11 13 15

A: The given problem is to solve the given congruence equation. 3(26!)= x (mod 29).

Q: 20. If a e Z and a = 1 (mod 5), then a² = 1 (mod 5). If a - h (mod n) then a3 = h3 (mod n)

A: Notation: a|b  means a divides b or                   a|b means b is divisible by a.

Q: For which primes p > 7 is 7 a square mod p? Your answer should be of the form p = {a1, a2,... , a}…

A: Given: p&gt;7 ,If 7 is a square mod p.

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

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