Let p be a prime number. Prove that for all nonzero a € Zp, z-a is a divisor of r²-1-1 in Zp[r]. Hence, we must have 2-1 p-1 -1= II(x-a) a=1 in Zp[r].

Q: Prove that 10n +9 and 6n + 5 are relatively prime for all n E Z.

A: Two numbers are said to be prime  to each other if they  share the gcd 1. That is to say  suppose…

Q: For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n +…

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Q: Prove the following statements by contradiction. Theorem. (a) There do not exist integers x and y…

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Q: Prove that if r <s < n, then P(n , s) is divisible by P(n,r).

A: Given: r≤s≤n Therefore, n-r≥n-s So we can write n-r! = n-rn-r-1n-r-2...n-sn-s-1n-s-2...1 n-r! =…

Q: Show that, for all constructible numbers y € R, x³ – 2 is irreducible over • Q(r).

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Q: Let d be an integer less than -1 that is not divisible by the square ofa prime. Prove that the only…

A: In this question, It is given that, d be an integer less than -1 that is not divisible by the square…

Q: Prove that o – 1 is divisible by 4.

A: Even numbers are numbers divisible by 2. Odd numbers are numbers that are not divisible by 2. The…

Q: Let p be a prime divisor of a positive integer n. Prove that p is irreduciblein Zn if and only if p2…

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Q: Prove that for every positive integer m, there is a positive integer k such that k does not divide m…

A: Given:-                 prove that every positive integer m, there is a positive integer K such that…

Q: Prove that there are no positive integers m and n such that √2m = πn.

A: We have to show that there are no positive integers m and n such that 2m=πn .

Q: Prove the following statements by contradiction. Theorem. (a) There do not exist integers x and y…

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Q: the set of all elements in Z, that are not zero fac- tors. An element a e Zn is a zero factor if…

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Q: Exercise 4.6. Prove that if h1, h2 are two greatest common divisors of f1, ..., fs, then there…

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Q: Prove that π2 -1 is algebraic over Q(π3).

A: We have to prove that π2-1 is algebraic over Qπ3.

Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: rove that if r < s < n, then P(n, s) is divisible by P(n,r).

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Q: Show that every integer of the form (6m + 1)(12m + 1)(18m + 1), where m E Z+ such 6m+1, 12m+1, and…

A: Carmichael number is an odd composite number which satisfies the  congruence relation  bn-1=1(mod n)…

Q: Let a be a non-zero element of Z43. w| 42. Show that [3] is a primitive root of Z43 and that [2] is…

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Q: Find all gcd’s of the two Gaussian integers 5 + 3i and −5 + i.

A: By repeated use of Euclidean division algorithm  we get the value of g.c.d.

Q: Prove that for all n in J, 1 + 3 + 5 + • • • + (2n - 1) = n2

A: We have to prove 1+3+5+....+2n-1=n2

Q: Let z be a root of z5 – 1 = 0 with z # 1. Compute %3D 215 + 216 + z17 +· . + z50

A: Since z is a root of z5-1=0 ⇒z-11+z+z2+z3+z4=0⇒1+z+z2+z3+z4=0…

Q: In Z[√ 5], prove that both 2 and 1 + √ 5 are irreducible but notprime.

A: Consider the norm function N(a+b√5)=a²−5b², which is multiplicative in Z[√5] , in the sense that…

Q: et a, b, c be non-zero integers. Suppose that x, y are integers such that ax + by = c. Show that for…

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Q: Prove that exists integers x, y, and z such that 1/x + 1/y + 1/z is also an integer.

A: Given: The expression is 1x+1y+1z where x, y and z are integers. To prove: There exist integers x,…

Q: Let p be a prime. Show that xpn −x+1 ∈ Fp[x] is irreducible iff either (1) p = 2 and n =1,2 or (2) p…

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Q: Give an outline for a proof of "for any integer r and any positive integer d, if d|(x– 1), then d…

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Q: Prove that π2 - 1 is algebraic over Q(π3).

A: In this question, we have to show that π2-1 is algebraic over ℚπ3. We know that any element α in…

Q: Prove that Z Q has the universal property of inverting non-zero elements.

A: in Mathematics, Z is the set of Integers. and Q is the set of all rational numbers. Z→Q Let f(α)=1α…

Q: Give a counterexample which proves the statement below is false over the doma integers. Vz((z – 20)…

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Q: Prove that (z) = (zk) for every integer k (provided z 0 when k is negative).

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Q: (a) Assume that n1, n2 and n3 are integers with gcd(n1, n2) = gcd(n2, n3) = gcd(n1, n3) = 1. Let…

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Q: Suppose that L and K are subfields of GF(pn). If L has ps elementsand K has pt elements, how many…

A: It is given that, L and K be the subfields of GFpn. If L has ps elements and K has pt elements.…