how that in the domain of integers of the form a+bsqrt(-17) the factorization 169=13*13=(4+3sqrt(-17))(4-3sqrt(-17)) demonstrates that unique factorization into primes fails in that domai

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

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Q: Prove by cases that 25x² + 35x is an even integer whenever 5x-7 is an integer. Note here that the…

A: Apply the condition for 5x-7 to be an integer case by case.

Q: Prove that 2n32n - 1 is always divisible by 17.

A:

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: Find the prime factorization of 793169. To show your work you must use Theorem 2

A: Consider the equation

Q: rove that the negative of any even integer is even

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Q: Use a direct proof to show that the product of two odd integers is odd.

A: Introduction: A direct proof is a series of statements, the final of which is the conclusion to be…

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: Use mathematical induction to prove that +1+2+4+...+2"-² = 2"- for all 2 -- positive integers . п.

A: We need to prove using mathematical induction given statement First we will show statement holds for…

Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

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Q: Prove that 5 does not divide 5n+3 for any integer n.

A:

Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) that 250 758…

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Q: Give a combinatorial interpretation of the coefficientof x4 in the expansion (1 + x + x2 + x3 +⋯)3.…

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: How do I find f prime of 3 if f(x)=13x/4+x2

A: Given,           fx=13x4+x2

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: How many integers greater than 5400 and less than 10000 have both of the following properties: -…

A: To count the number of positive integers satisfying all the required properties

Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

A:

Q: Show that x2 + 9 is prime.

A: To show x2 + 9 is prime.

Q: What is the largest number x such that x<200 and x has exactly 4 factors?

A: Factors:    Number that are multiplied together to get a product                              Factor…

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

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Q: The 4th term of an ap is zero. Prove that 25th term of the AP is three times its 11th term

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Q: Prove that n- + 1> 2" when n is a positive integer with 1 <n<4.

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Q: Show that there are infinitely many primes of the form 6k + 5.

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Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: Find ged(92928, 123552) and lem(92928, 123552) and verify that the ged(92928, 123552) x lem(92928,…

A: Prime factorization of 92928 is  92928=2×2×2×2×2×2×2×2×3×11×11=28×3×112 and…

Q: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: Find the canonical prime factorization of this using Fermat’s Method and solve for the Number of…

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Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

A:

Q: 3.) An airplane takes off and then climbs 2600 feet. After 25 minutes, it descends 150 feet. What is…

A: Please note that as per our guidelines if multiple questions are posted, we can answer the first…

Q: Can you please explain to me why is (k(k+1) ) / 2 is an integer and I know it always even so can you…

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Q: If x is an odd integer then 5x – 7 is an even integer. (use direct proof)

A: Given that “x” is an odd integer then “5x-7” is an even integer.

Q: is there integer that has reminder of 2 when its devided by 5?

A: There are infinite number of integers that has reminder of 2 when its divided by 5

Q: Find the GCD and LCM of the 2 214 and 25 125 using prime factorization

A: GCD=3 LCM=18542250

Q: Suppose that a, b. are two integers such that 3a² – b = 1. Does it follow that a and b are…

A: Given that,

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a non-negative integer

A: We need to prove by mathematical induction that, 3 divides 17k+1-2*5k+3+11k+3 whenever k is a…

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Q: how did they get 9x^2 from (x^4)/(5x^2)mod11 ?

A: Use of z11[x],

Q: Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In…

A: Introduction: Divisibility is a property of an integer. For two integers a and b with b&gt;a, the…

Q: 3. Prove that 0 divides an integer a if and only if a 0.(In your proof. you can freely ILNE your…

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Q: 4. Find them if: (a) lcm of two consecutive positive integers is 812; (b) lcm of twin primes is 899.

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

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Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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