Given four distinct prime numbers p, q, r, and s. It is possible to find numbers that satisfy the relationship n=p.q and n=r.s O Yes, this can happen if s | p.q O Yes, this happens if p = r + 1 and q = s - 1 O No, in such a case n will have four different factors O No, there is no case where such a condition would be met. O Yes, if p > r and q < s

Q: Let x and y be positive integers such that the prime factorization of x has exactly five 3's and the…

A: x = 35×k          k is an integery = 33×l           l is an integer

Q: If a and b are two distinct prime numbers, then the greatest common divisor of a and b is O ab

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Q: Prove the following: If m is an even number and n is an odd number, then n 2m is odd. Hint: start…

A: We prove that if m is an even number and n is an odd number , then n-2m is odd .

Q: (a) If p is a prime number and a € Z such that pla, determine whether pl(a + 1) (b) Show that the…

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Q: Every natural number n > 2 cam be written n = P1P2 Pr as the ... product of distinct prime numbers…

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Q: Let p be a prime number and a be a positive integer. Prove that p | (a^p - (p-2)!a).

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Q: 16.K. If c > 0 and n is a natural number, there exists a unique positive num- ber b such that b" : E…

A: Solution for 16K: If n=1, we can take y=x. So let n&gt;1. Let A=t∈ℝ:t&gt;0 and tn&lt;x Claim: A≠∅…

Q: 1. |x+y|≥x+y 2. 3=5, or 7 is a prime number. 3. If x is a prime number, then x is odd.

A: 1) For first statement we can show that by modulus of a number   2) One of the substatement is true…

Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

A: We have to solve given problem:

Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

A: twin prime numbers are the prime numbers which are separated by difference of 1 for example 11 and…

Q: Prove that if p is even and q is odd, then m+n -2 is odd

A: Value of p may be  2, 4, 6, 8, 10, 12, 14......... Value of q may be 1, 3, 5, 7, 9, 11, 13.......

Q: Which of the following is true? If p is prime and ab| p then a = 1 or b = 1 If lcm(a, b) = b then…

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Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: Replace the values of n by given values

Q: Given: "If x is an odd integer, then x + 2 is odd", what do you assume in a proof by contraposition?…

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Q: 1. Suppose that A = {r:x €N and r is even}, B = {r:x €N and r is prime}, C = {r :x €N and a is a…

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Q: Is the following statement true or false? For every integer n, if n is prime then (-1)^= -1. Which…

A: The given statement is: For every integer n, if n is prime then -1n=-1. We know that a non negative…

Q: 1-n 2+n D. For each natural number n, A, = (- "). Find UnEN An and Nnen An 2n

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Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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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: If n is prime, then 2" – 1 is prime.

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Q: Given E denote the set of even integers and O denote the set of odd integers. Let Z denote the set…

A: Given E=Set of even integers and O=set of odd integers And Z is the set of all integers. We have to…

Q: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

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Q: Which of the following correctly define what a prime number is? An integer p > 1 such that if d is…

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Q: 11. If p is a prime and p divides 4.":,, thenp divides " for some i.

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Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

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Q: If p is a prime number, then which of the following is true. Select one: (p – 3)! = 1(mod p) О (р-…

A: Let p be a prime number. We have to choose the correct option from the given list. Consider the…

Q: Let p be a prime number with p>=5. Prove that p must either be one more or one less than a multiple…

A: We use a fact that any number greater that 5 must be written as one of the form…

Q: integers a, b, and c with a≠0, if a∣b,  then a∣(bc). I need a know show table and a proof

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Q: Assume that the sum of the digits of a number, n, in base 10, equals to 16. Example: n= 97 is one…

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Q: If n is a product of twin primes, then a) Ø(n)o(n) = (n – 1)(n – 3) b) ø(n)o(n) = (n + 1)(n – 3) %3D…

A: If n is a product of twin primes, then a)  ϕ(n)σ(n) = (n+1)(n-3) b)  ϕ(n)σ(n) = (n+1)(n-3) c)…

Q: which of the following defines the positive integers divisible by both 3 and 4? Select one: O a. 12…

A: Explanation of the solution is given below...

Q: Let p be a prime number, n 2 2. Show that 6(pn) = po(n) if and only if p n. %3D

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Q: If A = {x : x is a natural number },               B = {x : x is an even natural number}  C = {x : x…

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Q: If a, bị c, d are distinct prime numbers with an as smallest prime the a * b * c *d is Prime number…

A: Given: a, b, c, d are distinct prime numbers and  a is the smallest prime number We know that, the…

Q: Evaluate |P(A)| if A={n: n is a prime number from 1 to 100}

A: To evaluate : PA if A=n : n is a prime number from 1 to 100

Q: 15/If n and n +2 are a pair of twin primes, establish that o (n +2) = 0 (n) +2; this also holds for…

A: σ (n) = sum of all the positive divisors of n

Q: For which natural numbers n is the following statement true? (3 X(n + 5n)) (n2 is irrational)

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

Q: 25. If p>1 is an integer and nłp for each integer n for which 2<n< p, then p is prime.

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Q: Suppose p and q are distinct prime numbers. Find the divide x. 1. number of positive integers r < pq…

A: Given that p and q are distinct prime numbers. To find: number positive integer x ≤pq such that…

Q: (a) Determine the set of values of n for which and are positive integers. A 'Pythagorean triple' is…

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Q: 1 of the above number n, or (65) = Note that the other number is Therefore, we conclude that the…

A: Given that, number is 15  of the given number n

Q: If p is a prime with p> 2, then 2p + 2 is a multiple of O a. 6 b. 4 Oc. None O d. 3 or 4 O e. 2 ar 5

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Q: positive

A: Given:       integers ±1 , ±2 , ±3, ±4Sum of four positive integer = + 10Sum of four Negative…

Q: 5) If p is an odd prime, then 2(p – 3)! = ?? (mod p) a) 2 b) 1 c) - 1 d) - 2

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Q: If p 2 5 is a prime, then p² + 2 1) prime 2) perfect integer 3) composite (Hint: L et n - 6L

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Q: Let N be an integer, and let p be a prime > 7. Suppose p| (N2+7). Show that p is of the form 1+ 7k,…

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Q: Given p: n is an odd integer q: n is an even integer. Which of the following statement cannot be…

A: Given statements p: n is an odd integer q: n is an odd integer  The answer can be found by verifying…

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.