For each of the following statements, indicate whether the statement is true 1. or false and justify your answer with a proof or a counterexample. (a) Let m, d € N and assume that d | m. If a = a + dZ is a square in Z/dZ, then ã = a +mZ is a square in Z/mZ. (b) Let a, p, q € N with p and q odd primes. If p = q (mod a) then (=) = ().

Q: 1. Show that the square root of 3 is irrational as follows: (a) If 3p² = q² for p, q E Z, show that…

A: Let us prove the above result.

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: If A = (; A-: )-..R. e M2x 2(R), find A" for any positive integer n.

A:

Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

Q: Find the maximum value of a, x, +a,x, + .+ax to the condition that x +x, +. + x =1 for given…

A: We need to find the maximum value of a1x1+a2x2+a3x3+...+anxn to the condition that…

Q: 2. ) For each statement, Determine if it is True. If it is true then prove it. If it is False then…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: 19 (e) a = 1575, b = -231 (a) 3710

A:

Q: . Show that p(X) = XP-1 +... +X +1 when p is prime.

A: For every integer n≥1, the nth cyclotomic polynomial Φn is the monic polynomial whose complex roots…

Q: Let A = {n EZ | n = 5r for some integer r} and B = {mEZ | m = 20s for some integer s}. Determine…

A:

Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: 2. Prove that võ ¢ Q. You may also use the following facts without proof. (1) Each rational number r…

A:

Q: For all n, d EZ where d + 0, d\n n = dk for some kEZ. Disprove: For all integers a, 6, and c, if…

A: I have disproved the result

Q: 44. (a) Let a, b, ceZ with (a, b) = 1 and a | bc. Prove that a c. (b) Provide a counterexample to…

A:

Q: 'ind integers s and t such that s · 35 +t · 78 = 1. Show your work.

A:

Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

A:

Q: It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2)…

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