(1) What is gcd(31212021 + 3, 31212021 - 3)? I am aware of 3121 is a prime number, and 2021 is divisible by 43 and 47. But I don't know what to do next

Q: 2. Find the root using Regula-Falsi Method a) x* - x - 10 = 0 Ans. x = -1.697 and 1.855

A: Follow the procedure given below.

Q: Is 3 -1 1 diagnalizable? Why?

A: Consider the given matrix 3-115. Let A=3-115. For an n×n matrix A to be diagonalizable, the number…

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

A:

Q: how do you find the following GCD? gcd(n, 2n) where n is a prime integer

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Q: Is it true that (3 x EN) (x²-3x +20 is a prime)? O True False

A: Is it true that ( there exists x in N)(x^2-3x+20  is a prime ) ?   True False

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A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: I was asked to prove that 5^(2n+1) + 4^(2n+1) is divisible by 9 for all n that are positive…

A: SOLUTION-

Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: In the expansion of (2x+1)n , the coefficient of the x2 term is 40n , where n is a positive integer.…

A: Given (2x+1)^n The solution is given below in the next step

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

A:

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

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A: Given,           fx=13x4+x2

Q: How can I quickly tell that x + 6x + 7 is prime?

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Q: How many terms are there in the expansion of (4.x + 7y)0.+ (4x- 7y)10 ?

A:

Q: What is the multiplicity of the factor 5 for the number 5°1001º11?

A: Given number is 551001011 . we have to find the multiplicity of the factor 5

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: What is the fifth term of the expansion of (x^2 + 1/x)^10?

A: What is the fifth term of the expansion of (x^2 + 1/x)^10?

Q: Suppose n is a natural number greater than 100. Alex and Jerry each compute the gcd(4n, 20n2 + 15)…

A: Given that n is a natural number greater than 100. Alex and Jerry computed the gcd4n,20n2+15 and got…

Q: 4. Prove by contradiction: VmeZ, 7m+4 is not divisible by 7

A: Contrary suppose that,   m is an integer such that 7 m+4 is divisible by 7. By using the definition…

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A:

Q: Do all four problems. 1. Prove that there exists a unique prime number of the form n2 – 4, where n…

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Q: 5. Solve the following problems: 5a. By using proof by cases method, prove that "For all odd…

A:

Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: Do you think it is worthwhile, or useful, to solve the puzzles proposed in Euclid's theorem of…

A: The objective is to solve the puzzles proposed in Euclid's theorem of primes There are infinitely…

Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

A:

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

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…

A:

Q: Problem B Show that 3. 4" + 51 is divisible by 3 and 9 for all positive integers n.

A: We need to show that 3·4n+51 is divisible by 3  by using the principle of mathematical induction…

Q: If n = 4k + 3, does 8 divide n2 - 1? (Assume that n and k represent integers.) O Yes, because n2 - 1…

A:

Q: in the expansion of (x^2-(1/2x))^5 ) what is the coefficient of x^4?

A: Given that  (x2-12x)5 Apply binomial theorem:a+bn=∑i=0nnian-ibiHere,a=x2,  b=-12x ∑i=055ix25-i-12xi

Q: what is the particular solution to the difterential equahion y's +e, subject to y l) = 4e %3D why…

A: Given differential equation is y'=-yx+ex we can write it as y'+yx=ex I.F. is I.F.=e∫1xdx=elogx=x…

Q: Consider the following statement. For every integer m, 7m + 4 is not divisible by 7. Construct a…

A: See the solution below.

Q: Prove that n2+n  is  divisible by 2, where nN.

A:

Q: What is the coefficient of x12y13 in the expansion of (2x+3y)25?

A:

Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

A:

Q: "For all integers n, n(n² – 1) is divisible by 6." (You may use quotient-remainder theorem and…

A: Let us consider, for all integers n a = n3 – n a = n (n2 – 1) a = n (n + 1)(n – 1)

Q: Answer the following. 1. Find x if 2x309 is divisible by 9 2 Find the canonical prime factorization…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: 7x =4 (mod 12).

A:

Q: What is (3a- 82² – 10a – 24) ( – 4)? O 3a2 + 4x + 3+ エー4 4 O 3x? + 4x + 6 O 3x? + 4x + 3- -4 O 322…

A:

Q: how do you compute 1.2193 to the 1/3 power?

A: Here we use logarithmic method to find tha solution.

Q: If n is an even integer, then 3n° is an even integer.

A:

Q: 8. Show that no prime number of the form 4k +3 can divide a number of the form n2 + 1.

A:

Q: Compute the exact answer mentally by finding compatible numbers or substitutions: 4x 273 x 25 4x273…

A: The steps are as follows

Q: 2. 6n – 1 is divisible by 5 for n any positive integer

A:

Q: -36618)-5/-345 31-し -3+5 3/2)-5/2 )" -3(2xar2) -5(axaxa -3(8)-5(8) -24+-40 -64

A: To solve

Q: Show that, given a 7-digit number, where all digits except perhaps the last is non-zero, you can…

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Q: How do I find f prime of 13x/4+x2

A:

Q: For the previous problem how did you calculate for 1,5,17,and 37 for proving that n^2 + 1 is not…

A: Solution

Q: How can I quickly tell that x2 + 6x + 7 is prime?

A: a prime quadratic equation can not be factored or it can not have rational roots.

Q: Prove that n(n + 1)(n + 2) is always divisible by 6

A: