Prove that if p is a prime integer, then 3√p is irrational.

Q: Prove or disprove that if a, b E Q are rational numbers, then ab is also rational.

A: Statement: If a and b are rational number then ab is also rational.

Q: shoew that r+E and  rE are irrational.

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Q: (a) Prove that √ 95 is irrational. (b) Show that √ 19 − √ 5 is irrational.

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Q: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

Q: Use proof by contradiction to show that, "The sum(s) of an irrational number(i) and a rational…

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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: ) Give the contrapositive of the following statement concerning a positive integer p: If p > 2 and p…

A: Contrapositive of a conditional statement of form  (p^q)→r is  ~r→(~pv~q)

Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

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Q: Prove that if n is a positive integer, then 133 divides 11"+1 + 122n-1

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Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: Prove that Vp+1-\p is irrational where p is prime integer.

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Q: Prove that p is irrational for any prime p

A: To prove that √p is an irrational number.

Q: For which prime p,  p^2+2 is also prime?

A: Put p=3 in p2+2 and obtain that, 32+2=9+2=11. Clearly, 11 is a prime number. Thus, p=3 satisfies…

Q: Which of the following is not true? a) for any prime p, Jp is irrational. b) if Va is rational, then…

A: a.) To prove: For any prime p, p is irrational number. Proof: Let us proof this result by…

Q: 1. If today is Saturday, what day of the week will it be 1,500 days from today? 2. Prove that 4/2 +…

A: The explanation of the solution as follows:

Q: Prove that, if a is an irrational number, then Va is also an irrational number.

A: In the given problem we have to prove that if x is an irrational number, then x3 is also an…

Q: Prove that for each irrational number r, there exists a unique integer n such that |r – n| < 1/2.

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Q: 3. Prove that p is irrational for any prime p.

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the third question.  Please…

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

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Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

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Q: Prove that if p is a prime, then the square root of p is irrational.

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Q: Prove that if a is an irrational number, then a-1 is an irrational number.

A: Given:           a is an irrational number To prove:            a-1 is an irrational number

Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: If k is a positive integer, prove the k root of 6 is irrational

A: Given:Given that k is apositive integer.To find:The kth root of 6 is Irrational.

Q: Prove that if n is an integer and 5n + 6 is even, then n is even.

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Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

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Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &amp;  b/c  , we know there exists integers K1 &amp; K2  ,with K1 a=K2 b=c--1Also…

Q: Prove that a positive integer n is prime number.if no prime p less than or equal to n divides n.

A: Let us assume that the contradiction of the statement , let  n ≥ 0 be a composite number .  Thus , n…

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Prove that every prime p ≥ 5 is of the form p = √(24k + 1 )for some integer k.

A: To Prove that every prime p ≥ 5 is of the form p =√(24k + 1 )for some integer k. Or equivalently, p2…

Q: Number theory.

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Q: Show that square root P is irrational where P is prime by using contradiction.

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Q: Give a proof to the following: For all integers aa and bb, if a|13a|13 and 13|b13|b, then a|ba|b.

A: If, a | 13, then, there exists an integer s such that, 13 = a•s  And, if 13 | b, then there exists…

Q: Is the following proposition true or false? Justify your conclusion with a proof or a…

A: Check the following proposition, "For each natural number n, if 3 does not divide (n2+2) then n is…

Q: Prove that if m and n are integers and mnis even, thenm is even or n is even.

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Q: a. If p E P is a prime number, prove that the cube root of 2p is irrational.

A: We will use the basic knowledge of number theory and logic to answer this question. Note that only…

Q: А. If p 2q 5 is a prime number, prove that p2-q? is divisible by 3.

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Q: Prove by contradiction: V5 is an irrational number.

A: Let 5 is rational number ,  therefore, it will of the form pq  now 5=pq where p and q are both…

Q: Prove that Let p be a prime. Then (p−1)! ≡ −1 (p)

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Q: Prove that for any prime positive integers P,VP is an irrational number

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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: If n is a perfect number, prove Σ d|n 1 2.

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Q: If x is a rational number and y is irrational, prove that x +y and x y are irrational.

A: There are many methods to prove a statement. Some of them are direct proof, prof by contradiction,…

Q: . Prove that V6 is irrational number. -. If a, b e R, prove that ab <(a² + b°).

A: We have to prove that 6 is irrational number

Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: Prove that, if z is an irrational number, then Va is also an irrational number.

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Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.