Show that the number of binary operations on {1, 2} having 1 as identity and having 2 as the inverse of 2 is exactly one.

Q: determine whether multiplication by A is one-to- one by computing the nullity of A and then applying…

A: Let (x, y) be in nullspace of A.  Then, A(x, y)t = 1-22-4-36xy = 000     => x - 2y = 0…

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

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: Prove that 2n32n - 1 is always divisible by 17.

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108.

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: O Show that Z4 (under addition) is isomorphic to Z% (under multiplication

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Q: Find all solutions of the linear congruence: 9r = 5 mod (14) Define a modular inverse. Find a…

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Q: Show that addition and multiplication are associative binary operation onR. But subtraction is not…

A: Given operations addition, subtraction, multiplication and division.

Q: Suppose that an < bn < Cn for all n, an → L, and cn → L. Show that bn → L.

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Q: Prove that (1234) is not the product of 3-cycles.  Generalize.

A: Given: Given cycle is 1234 we want to prove that 1234 is not product of 3-cycles.

Q: What are the two things we need to show to prove that need to show to prove that is a binary…

A: To find: the conditions such that "*" is a binary operation or binary composition.

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

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Q: Compute the pseudo inverse of A

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Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: Find the multiplicative inverse of the given element   [9] in Z242

A: To find the multiplicative inverse of the given element 9 in Z242.

Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: Determine which elements of Z8 have inverses under the operation ×8. Then do the same for Z7 under…

A: For an element a a multiplicative inverse is an element b such that ab=1. The set Zn is the set of…

Q: Find inverse of 55 modulo 243

A: Find inverse of 55 modulo 243

Q: 4. Is the binary operation # defined on the set of real numbers commutative if a#b = a+ 2b + 4, Va,…

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Q: Determine which elements of Z8 have inverses under the op- eration ×8. Then do the same for Z7 under…

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

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Q: Show that (AB)¹ = BTA¹ if A = 2 0 2 0 and B = -1 1 15) -3 1 -1 -1 2 2). a

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Q: Show that if x is an integer, then gcd(x + 1, x2 – x + 1) divides 3. -

A: First, we find gcdx+1, x2-x+1 Now, factors of x+1 are x+1, 1 x2-x+1 is a quadratic equation of the…

Q: Exercise 14.6.10. Prove that it is impossible to write the identity permu- tation as the product of…

A: We have to show it is impossible to write the identity permutation as the product of an odd number…

Q: Prove that the product of two odd numbers is odd.

A: We first define a general odd number. Lets say that n is an integer. When we multiply any number by…

Q: ab Q4. Consider the binary operation * on I-, the set of positive integers defined by a * b = 2 %3D…

A: It is given that the set of positive integer I+=1,2,3,... and Given operation a*b=ab2. Binary…

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Prove that 4.999...=5.000...=5

A: Proof:Let x = 4.999, then 10x = 49.99.

Q: is The inverse element with respect to the binary operation • defined by x • y=:

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Q: 15-1mod 17 (use Fermat theorem)

A: We will answer the first question as we don't answer multiple questions at a time. Please resubmit…

Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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Q: Every natural number has a successor every number except 1 has a predecessor

A: We know that  Successor  is that number which can be find adding 1 in the given number  So if we…

Q: Find inverse of 65 modulo 243

A: The definition of inverse modulo is: a is said to be the inverse of b modulo p, if ab≡1(mod p)

Q: Find [7]^-1 in Z55 under multiplication.

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Q: Please check image

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Q: 3. Prove that eª >1+x for x > -1

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Q: which of the following is a binary operation if a and b are in S

A: To find:  Which of the given is a binary operation if a and b are in S

Q: Prove that for а + —1,а > 0, пENis (1+ а)" > 1+ па.

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Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

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Q: compute the following in Z12: [9]+[11], [5]-[8], and [7][9].

A: Consider the provided question, We have to find the following in z12:9+11, 5-8, and 79 Find,…

Q: Find a multiplicative inverse of  21 modulo 46

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Q: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

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Q: Show that, given a 7-digit number, where all digits except perhaps the last is non-zero, you can…

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Q: The inverse with respect to the binary operation on R defined by x • y=-

A: Given; x ° y= (x+y)/6

Q: Find the integer a, if the modular representation of a is (2, 1, 5) with respect to the modulo 3,…

A: We have to find the integer a if the modular representation of a is 2,1,5 with respect to modulo 3,…

Q: Prove that ifx^2−6x+ 3is odd, then x is even

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Q: Show that addition, subtraction and multiplication are binary operationson R, but division is not a…

A: The given relation is R.

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…