a) Find the number of positive integers not exceeding 120 that are divisible by 3 or 4. 1 0 01 0 0 0 1 b) Let A =|1 01. Find A[9]. 0 0 0 0 1 1 -0- 0-

Q: Abstract Algebra. Please explain everything in detail.

A: To prove (or disprove)  the possibilty of required representations under the given conditions

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A:

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

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Q: a) Prove that, for all integers K > 5 Р(к +1,5) — Р(К,5) %3D 5P (К, 4)

A:

Q: 1.(a) Do there exist integers a, b, c such that gcd(a, b) = 30, gcd(b, c) = 24, gcd(a, c) = 54?…

A: (a) gcd(a, b) = 30       gcd(b, c) = 24       gcd(a, c) = 54      =&gt; 30 | a, 54 | a           24…

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A: answer

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A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: 5. Let a be a real number, c an integer, c > 0. Prove that

A:

Q: Show that if a, b, c, and d are integers with a and c nonzero, such that a|b and c|d, then ac|bd.

A:

Q: Abstract Algebra

A: To find non-negative representations under the given conditions

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A: Solution is in step 2

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Q: Let a and b be two integers. Prove that if a+b is even then a-b is even.

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A: using :- if a|b then a=bq , where q is unique integer.

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Q: 4. Show that if x, y, z are integers such that x2 + y2 + z? = 2xyz, then x = y = z = 0. (H)

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Q: Prove, directly from the definition of divisibility, that for any integer x#0, x|0, 1|x and x|x.

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A:

Q: Prove that every integer greater than 27 can be written as 5a + 8b, where a, b e Z+.

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A: Solution

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Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.