Given positive odd integer (a), find all possible divisors, which are always divided a + (a + 1)° + (a + 4)³ + 3

Q: (a) Show that n(n2 – 1)(n + 2) is divisible by 4 for all integer n.

A: As per guidelines , if there are more than one questions , we have to solve the first question only…

Q: Let N=34 * 34* 63* 270. What is the ratio of the sum of the odd divisors of N to the sum of the even…

A: The number N is calculated by the prime…

Q: Find the greatest natural number n such that n + 10 divides nº + 100.

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Q: 8) The number 4050 can be factored into prime divisors as: 2 x 3 x 3 x 3 x 3 x 5 x 5 a) Determine…

A: We have to solve

Q: the fourth power of an integer is x, and x is between 100 and 600. What is x?

A: It is given that the fourth power of an integer is x.That is, x = n^4, where n is a positive…

Q: Determine the coefficient of a° in (1– 5a³)® by completing the following given steps.

A: we have to determine the coefficient of a3 in 1-5a35. as we know that if we have (x+y)n then the…

Q: a. Use your values of a, b and c to find the term independent of x in the expansion of 5 1 (2 + 9x)2…

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Q: Question 5 Find all integers n >1 such that 1" + 27+...+(n – 1)" is divisible by n. |

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n Proof: Let…

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: Show that there exists i and j with i + j such that p, divides p;. а. b. Show that there exists i…

A: Given: It is given that let p1,p2,...., p51 be their final exam scores . Hence each pi is an integer…

Q: Find all four-digit numbers abba with the property aa · 10b = abba if a

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Q: P(x) = 5 x + 13 x5 − 7 x6 − 2 x2     If P is written in descending order, what would be the final…

A: given polynomial is                                               P(x)=5x+13x5-7x6-2x2 P is written…

Q: Construct the multiplication table for Z5 Using this table, or otherwise, determine the following x…

A: Solution

Q: what is the element a34

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Q: Question 1 Find all integers n>1 such that 1" + 2"+...+(n – 1)" is divisible by n.

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Q: Show that (a) 8n −3n is divisible by 5 for all natural numbers n.  (b) n2 −n +41 is odd for all…

A: expansion of (a+b)n = hco an bo + nc1 an-1 b1 +.......... +ncn ao b n

Q: Given two integers n and m such that their remainders after division by 5 equal to 1. What can be…

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Q: If n is a natural number, the number 34n+4 – 43n+3 is divisible by: -

A: Given : To find : A positive number which can divide above expression

Q: The greatest common divisor of a and a +5 is a. 5 Ob None O. 1 or a ed. l or 5 Oe. 1

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Q: Vn E N, nº – n is divisible by 6.

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Q: find all possible three-digit numbers that are of the form 75_ and are divisible by both 3 and 2

A: Since the number is divisible by 2 and 3, it must be divisible by 6.

Q: For each natural number n, the expression 23n – 7n – 1 is divisible by

A: The binomial theorem is given by  (x+a)n=C0n xn a0+C1n xn-1 a1+...+Cnn x0 an…

Q: B. Using induction, prove that 8" -1 is divisible by 7, where ne N

A: (B) 8n-1 is divisible by 7 Now for n=1 81-17=1 For n=2 82-17=64-17=637=9 Now, assume that 8n-1 is…

Q: For any integer a, show that a²-a+7 ends in one of the digits 3, 7, or 9.

A: For any integer a , we have to show that a2 - a + 7 ends in one of the digits 3 , 7 , or 9 . Note…

Q: (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

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Q: Find an expression, in terms of n for the coefficient of x in the expansion (1+4x) + (1+4x)² +…

A: ANSWER :

Q: 5. Given a real number x and a positive integer k, how many multiplications are used to find kx by…

A: Given,A real number x and positive integer kTo find: Total number of multiplications used with the…

Q: If x is odd, then (x^2+3)(x^2+7) is divisible by 32

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Q: Find an integer that leaves a remainder of 2 whan divided by either 3 or 5, but that is divisible by…

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Q: Find the term of (a – 2b)3 that has b in it. |

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Q: For n ≥ 2, let an=5n/n2-2 Determine for which n, |an|<1

A: Use the given inequality for an and find the quadratic equation for n and solve

Q: For n≥2, let an= 5n/n2-2. Determine for which n, |an|<1.

A: Use the inequality on and find the inequality for n

Q: Show that a6-1 is divisible by 168 whenever (a, 42)=1.

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Q: Infer a formula for Ak using factorization where A =

A: Calculate eigenvalues: 1-λ-62-6-λ=0-6+6λ-λ+λ2+12=0λ2+5λ+6=0λ=-2,-3

Q: Find all positive integers n for which n1º +1 is divisible by 10.

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Q: In a round-robin chess tournament, each player is paired with every other player once. The formula 2…

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Q: 5. Find the value of AB. B AL 5 in 5 in X16 in

A: We have to find AB

Q: If n is an integer greater than 1, then (a + g) - C, (a - b +2²g) + C2 (a - 2b +3² g) +(-1)" (a - nb…

A: Given n&gt;1, a+g-C1a-b+22g+C2a-2b+32g-.....+-1na-nb+n+12gCn…

Q: How many terms will be in the expansion of (2x+3y)^6?

A: How many terms will be in the expansion of (2x+3y)^6?   Use the binomial expansion theorem to find…

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

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Q: Let N = 34 · 34 · 63 - 270. What is the ratio of the sum of the odd divisors of N to the sum of the…

A: We have the number N given by N=34·34·63·270. From the basics learned in the earlier stages, we have…

Q: The Greatest Common Divisor of two prime numbers a and b

A: Prime numbers are those numbers which have only two factors one factor is 1 and other factor is the…

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: Write an explicit formula for the recur an = a(n-1) - 0.8; a = 1.6 )(n – 1) an

A: So in this we have to apply the standards to change into explicit form, I'm providing hand written…

Q: Let -6 M = Find formulas for the entries of M", where n is a positive integer.

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Q: Suppose you have 13 Physics texts, 7Computer Science texts, and 9 Math texts. How many different…

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Q: Prime divisors of two numbers a and b are 2,3 and 5 and product of these numbers a and b is 5400.…

A: The prime factors of the two numbers are 2, 3, and 5 and the product of these two numbers is 5400.

Q: Find the sum of the divisors of 120. Generalise the result for an arbitrary natural number M

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Q: A box with a square base and no top is to be made from a square piece of carboard by cutting 2 in.…

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Q: Abel says (6)- 6°. Is he correct? Explain.

A: Given statement  (6^3)^3=6^5 Is the given statement correct or wrong.