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

Q: 13. If a is an algebraic integer satisfying a + a + 1 = 0 and ß is an algebraic integer satisfying…

A: Given: α is an algebraic integer satisfying α3+α+1=0 and ß is an algebraic integer satisfying…

Q: 23. For any real number a, a ‡ 0, show that a +1 a +4 a +7 a +5 a+6 a +9 a -2 a +3 and a +8 | = 0, a…

A:

Q: X7. We know that given any real numbers x1 and x2, if x1x2 = 0, then x1 = 0 or x2 = 0. Since there…

A: (b) given any real number x1 , x2 , x3 , x4 claim- if x1x2x3x4=0 then x1=0 , x2=0 , x3=0 , x4=0

Q: For any real number a, a ≠ 0, show that a+4 a+7 a+5 a +8 = 0, a+6 а+9 a+1 a+2 a+3 and 4a7a 5a 8a =…

A: We are required to prove: a+1a+4a+7a+2a+5a+8a+3a+6a+9=0a4a7a2a5a8a3a6a9a=0aa4a7a2a5a8a3a6a9=0

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

A:

Q: Show that the square of any positive Integer is of the form 3m or 3m+1 for some integer m

A:

Q: For any integer z, if 3 divides z2, then 3 also divides z.

A:

Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

A:

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A: we have 15a+20b =49 and a and b both are in R  we prove that a and b are not both integers.

Q: Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing…

A: Since you have asked multiple questions, we will solve the 1st question for you. If you want any…

Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

A:

Q: What is the element a32 from A1 1 - 2 - 1 - when A 2 1 1 -1 2 - 3

A:

Q: MUI QUE If a and b are positive integers, which of the following can never be the gcd(56a, 63b)? (a)…

A:

Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

A:

Q: Assume that c and d are particular integers. Is 6c + 4d ² + 3 odd? 2

A:

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Z/mZz Zm

A: We need to prove that Z / mZ is isomorphic to Zm .

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A:

Q: where there are a; + 1 options for each exponent x; because 0 ≤ x ≤ a,, and so the number of…

A:

Q: Using Fermat’s little theorem to show that 42 | n7 −n, where n is a positive integer.

A:

Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Give a non-zero numerical counterexample to the following statement: Every multiple of 12 is also a…

A:

Q: Theorem: The sum of any even integer and any odd integer is odd. Construct a proof for the theorem…

A: We have to prove the theorem by scrambling the given statements.

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: ndicate the least integer n for which (3x5 + x2)4 is O(xn).

A: Solved using the binomial theorem

Q: take any 11 integers. prove that there must be at least two of them that differ by an integer…

A:

Q: Which is a factor of both x2 + 7x + 12 and – 3x – 12? O r - 4 O r- 3 O x + 4 O x +3

A:

Q: (c) Prove that x - 11 |x| + 11 |x² + 36| 36 for all real r.

A:

Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.