Show that if n = p1p2⋯pk, where p1, p2,…, pk are distinct primes that satisfy pj − 1 | n − 1 for j = 1, 2,…, k, then n is a Carmichael number.

Q: 1. Let n be a positive integer and le {1,...,n} such that n divides a, (ai+1-1) for i =

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Q: 2. Let N = 210. Prove that N-p is prime for every prime p such that N/2 < p < N. Find a prime number…

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Q: 4. There are 51 software students taking MATH 211. Let p,Pz, .. Psi be their final exam scores.…

A: 4. Given that,There are 51 software students taking MATH  211.Let, p1, p2, ..., p51 be their final…

Q: Let A be the set of irrational numbers in [0, 1] and let B be the set of ratinal numbers in [0, 1].…

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Q: Let p be a prime. Show that there is exactly two elements a belongs to Zp s.t a^2 = 1.

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Q: 3. If π is a prime in Z[i], then the number of elements in Z[i], is N(π). 4. If p, q are positive…

A: Let's address each statement one by one:3. If π is a prime in Z[i], then the number of elements in…

Q: 3.5. Let n E N. Prove that if n + ÷ < 2, then n² + n2 <4.

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Q: Let n be a positive integer and p a prime that divides n. Prove that p is prime in Zn.

A: As per the question we are given a integer n and one of its prime divisor p Now we have to prove…

Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

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Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Let n=pp Z+ where p₁ and p2 are distinct prime numbers and 7₁,72 ≥ 1. If o(n) = 80, then determine…

A: Let r1 and r2 be 1.So, n = p1 x p2The totient function, Φ (n) for any value of n is the count of…

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 6. Let p e Z be prime and n e z+. Prove that the largest power of p which divides n! is given by…

A: In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a…

Q: Prove that pork = pur in O₂ if and only if 0 w = 2m for some m€ Z and k = 1 (mod 2)

A: To prove that ρθrk=ρωrl in Ο2 if and only if θ-ω=2πm for some m∈ℤ  and k¯≡ l¯mod 2.

Q: * 30. Let n and k be integers with 1 ≤ k ≤n. Show that n Σ() ( k=1 k n - 1) = (2n+2)/2 - (2n).

A: The answer for the given questions has been solve step by step below please find the attachments…

Q: Let m 2q be an even positive integer. Prove that every integer is congruent modulo m to a unique…

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Q: 1. Compute the least nonnegative residue of 10k + 1 modulo 13 for k 1, 2, 3, 4.

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Q: 5. Let n be a positive integer and p1,P2,P3, . „Pn be n distinct primes. If m is a positive integer…

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Q: Assume that 0  ≤ a1 ≤ 1 and that  an+1=3√(1+an)/2 for all n ≥ 1. Prove that (a)0  ≤ an ≤ 1 for all…

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

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Q: 6. Let pe Z be prime and n E Zt. Prove that the largest power of p which divides n! is given by…

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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: If mand n are positive inte gers with m<n and p is a prime, then pmZpris isomor phic to Z p and F…

A: If m and n are positive integers with m&lt;n and p is a prime, then we have to show that is…

Q: Let m%3= i=1 where pi 0 are integers then i=1 i=1 gcd(m, n) = I min(ai,bi). %3D i=1

A: “Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: How many positive integers n less than or equal to 11,347 satisfy d(n) 1 (mod 2)?

A: We need to find a number of positive integers n≤11347 such that d(n)≡1(mod 2) i.e., , k∈Z.So, we…

Q: Prove that for any natural number n, 2(√√n +1 − 1) <1+ 1 √2 + <2√n.

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Q: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: n² < 2" for n = 4,5, 6, ... (Hint: 2n + 1 < 2" for n = 3,4, 5, ...)

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Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

A: Given: p≡1mod4       ∵p=4k+1

Q: Let a and b be odd positive integers. If there exist integers s and t with as+bt=4 then * O…

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Q: Let c be an element of N, and p be a prime. Prove that if c is greater than or equal to 3, then c^p…

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Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: n + 5 > 4 n

A: We know, the square of an integer is always non-negative (n - 2)² ≥ 0 for all integers n

Q: Let m = p₁¹p²²ª...p"² and n = p₁¹p2...pt where p₁,..., Ps are distinct prime numbers ans ri, ti ≥ 0…

A: Given that  and , where  are distinct prime numbers and  for all .We need to prove that:, where  for…

Q: (14) Let E denote the set of even integers, O denote the set of odd integers, and x, y € Z be any…

A: I have given the answer in the next step. Hope you understand that

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Let p be an odd prime and a, b are integers with pła, płb. Show that ab ()-(()) = Р

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Q: 3) Let II" , m i=1 where p1 0 integers. Show that d eN is a divisor of m if and only d = II p§ª…

A: Prove the theorem.

Q: Let k> 1 be such that p = 6k+1, q = 12k + 1, and r = 18k+ 1 are prime. Show that n = pqr is a…

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Q: 6.) Prove by induction that 1/5 + 1/3/53 13 + *** ㅗ 402-1 n for every nz | 2n+1

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Q: 6. If |G| = pq for distinct primes p and q, show that G is not simple. (Note the slight difference…

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Q: 18 Show that if p is a positive integer such that both p and p² + 2 are prime, then p= 3.

A: We have to prove that if p is a integer such that both p and p^2+2 are prime , then p=3.

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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