Use Euclid's Algorithm to show that there are integers s and t such that 21s + 97t 1. Showing all your working, compute [21]¬1 in Z97. How many elements of Z97 have a multiplicative inverse? Compute [4]-1 o [10] in Z13.

Q: Find the number of elements in the conjugacy class of (abc)(def) in S₆

A: We have to find the number of elements in the conjugacy class of (a b c)(d e f) in S₆

Q: Use the factorization A = PDP Ak= - 7 40 - 2 11 = 45 1 1 1 01 30 to compute A I -1 5 1 - 4 1 where k…

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Q: Let n be an integer. Prove that GCD(n² n + 1, n + 1) equals 1 or 3. Show by examples that both…

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Q: Use direct proof to show, "If m + n and n + p are even integers (s and t), where m, n, and pare…

A: To answer the given questions we shall use the following definition  An integer n is said to be even…

Q: Compute for an arbitrary integer k

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Q: Find the order of all element of Z8 under addition mod . Please brieflt explain and show all the…

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Q: Prove that 5n-1 is divisible by 4 to any positive integer n

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Q: a

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Q: Apply the Extended Euclidean Algorithm to find the multiplicative inverse of a mod n, provided it…

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

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Q: Find the least residue of  -5 -46 modulo 14.

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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: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

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Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

A: Given: Let n be a positive integer such that 78n-1. To prove: 78n+1-1

Q: A binary string is a word containing only Os and 1s. In a binary string, a 1-run is a non-extendable…

A: def count_1_runs(n): binary_representation = bin(n)[2:] # get the binary representation of n…

Q: x² + 4x = 10(mod11) (mod11) X=

A: Given that x2+4x≡10 (mod 11)x2+4x+4≡10+4 (mod 11)x2+4x+4≡14 (mod 11)x2+4x)(x+2)2≡14+11 (mod…

Q: Show that there are no integers a, b such that a² + 2ab+26² = 100000000003. Hint. Use mod 4…

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Q: 2. When the expression (156)(205) (557) is evaluated, it ends with a string of consecutive zero's.…

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Q: Select the following statements that are true. gcd (2022, 111) = 1 2023 is a prime gcd (2022, 1685)…

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Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: 3. Use induction to prove that 15"+2 + 162n+1 is divisible by 241 for every nonnegative integer п.

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Q: Suppose 59 cars start at a car race. In how many ways can the top 3 cars finish the race? ... The…

A: There is total 59 cars.

Q: 4. Does 3 divide (3k + 1)(3k + 2)(3k + 3)?

A: We will show by induction, put the value of k = 0 we find, (3x0+1)(3x0+2)(3x0+3) = 6  Which is…

Q: Z+ as follows: For a, b € Z+, a*b = a - b + 1. Z+? Explain. (3.1) Define an operation * on Is * a…

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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 <a21>∩<a10> .

Q: Q3. By using mathematical induction, show that 52n – 1 is divisible by 24, for all positive |…

A: We need to show that 52n-1 is divisible by 24. We can solve this using induction method.

Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

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Q: If a and b are positive integers, which of the following can never be the gcd(56a, 64b)? (а) 56а (b)…

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Q: Suppose that a, b are integers such that 1 = -19a + 45b. Use this information to find all integers a…

A: The given equation 1=−19a+45b suggests that 1 is a linear comb.−19 and 45 with coefficients a and…

Q: Show that the system consisting of all elements of the form a+b√5, where a and b are any rational…

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Q: Using Euclidean Algorithm, find the Inverse of the integer 13 mod 19.

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Q: Give a recursive definition for the set of all the positive integers that are less than 100 and not…

A: Given: Set of all positive integers that are less than 100. To find: Recursive definition for the…

Q: Suppose a is divisible by b. Show that gcd(a, b) = b.

A: Consider the given information. As we have given that a is divisible by b. then there is a number c…

Q: Prove that 3a^1 is even if and only if a is an odd integer.

A: In the given question we have to prove that Prove 3a^1 is even if and only if a is an odd integer.