Let p be a prime number. Prove that 1 and p − 1 are the only elements of Zp that are their own multiplicative inverses.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A: n is an integer. To show: 2n3 + 4 is divisible by 4 then n is even.

Q: Prove that for any integer a, 3 divides

A: To prove that for any integer a, 3 divides (a-3)(a+4)(a+2).

Q: Prove that, nonetheless, for all integers a, if 2 | a and 3 | a, then 6 | a

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Q: Prove that for any integer n, n6k - 1 is divisible by 7 if gcd(n, 7) = 1 and k is a positive…

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Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

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Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Use the Chinese remainder theorem to show that an integer a, with 0 ≤ a < m = m1m2⋯mn, where the…

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Q: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

Q: Suppose p is prime and a is a non-zero element of Z,. Prove that xP – x + a has no zero in Z,

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Q: 11. Prove that if d divides a, then dk divides ak for every positive integer k.

A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

Q: Prove that for all positive integers n, 4n-1 is divisible by 3.

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Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

Q: 3. Show that if a and b are integers such that alb, then ak|b* for every positive integer k.

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Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

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Q: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

A: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

Q: Prove that for any integer a , one of the integers a,a+2,a+4 is divisible by 3

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Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

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

A: Let P(n) be the statement, "For n∈ℕ there exist a, b>0 such that              n+27=a·5+b·8 Use…

Q: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

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Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Prove that if ni is a positive integer, a± 0

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Q: Prove by induction that for every integer n 2 1, (xe2*) = 2"-1(2x + n)e", ne Z

A: We prove this by Mathematical induction

Q: z is a subring of of Q: O True O False

A: Given statement is Z is a subring of Q.  We know that N is a subring of Z, which is a subring of Q,…

Q: Prove that if ∣A∣ = 1 and all entries of A are integers, then all entries of ∣A−1∣ must also be…

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Q: Prove that a positive integer n is p

A: Topic :- Number system

Q: Prove: If Am = O for some positive integer m, then I − A is nonsingular.

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Q: Suppose a, b, and c are integers and suppose a |b and ałc, where a{c denotes that c is not divisible…

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Q: Suppose a is a positive integer and p is a prime. Prove that pla if and only if the prime…

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Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Prove that if a > 0, k > 1, and ak -1 is prime, then a = 2 and k is prime. %3D

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Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A: Fermat's Theorem : If p is any prime and a is any integer then p divides (ap-1-1).

Q: Suppose that j is prime and j2 | ab. Prove that if gcd(a, b) = 1, then j2 | a or p2 | b.

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Q: (a) Let a,b, and c be integers such that a | b and a | c, and let x and y be arbitrary integers.…

A: According to the guidelines I am answering first question only. a) a, b and c be integers such that…

Q: Prove for every pair p, q of distinct primes that

A: To prove that for every pair p, q of distinct primes that pq is irrational.

Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: Vn € Zt [obeb(n, n + 1) = 1]

A: l  let n,n+1 are two consecutive positive integers let  the greatest common divisor i.e…

Q: 6. Prove that for all integers a, b, and c, if a² + b² = c², then at least one of a and b is even.

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Q: division of a2 with 4 is either 1 or 0.

A: We divide integers into 4 different class. Then work on each class of numbers.

Q: .3. a. Prove that for every integer a, if a' is even then a is even.

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Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

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Q: Prove that if D is a PID and p ∈ D is irreducible, then p is a prime of D

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Q: Let p1, p2, p3 denote three distinct prime numbers, and set n = P1P2P3. Show that if a congruence…

A: As per the question we have to show that the non-invertible & non-zero congruence class [a]n…