(a) Suppose that there are k distinct positive integers that divide an odd positive integer n. distinct positive integers divide 2n? How many divide 4n? (b) Suppose that there are k distinct positive integers that divide a positive integer n that is not divisible by 3. How many distinct positive integers divide 3n? How many divide 9n? How many

Q: Prove that N × N × N is countable. Does your argument generalize to the Cartesian product of k…

A:

Q: Prove that if n>1 is odd, then 1n+2n+3n+…+(n−1)n is divisible by n

A: by using mathematical induction, the given expression is proved and given below...

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

A:

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

A:

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

A:

Q: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

Q: 2. Prove that the number of irreducible fractions not greater than 1 and with denominator equal to n…

A: For the solution of the problem follow the next step.

Q: Let n be an integer. Prove that if 2n +4 is divisible by 4, then n is even.

A:

Q: An integer n > 2 is composite if and only if n does not divide (n - 1)! + 1. :Select one True False

A: The solution is given as

Q: Prove that 5n-1 is divisible by 4 to any positive integer n

A:

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

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

Q: divisible by 3 for

A:

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: Show that if the statement 1 + 2 + 22 + · · · + 2n - 1 = 2n is assumed to be true for n = k, then it…

A: Given, 1 + 2 + 22 + · · · + 2n - 1 = 2n ------------(1) is assumed to be true for n = k To prove -…

Q: Show that for every positive

A:

Q: Prove that n3 + 2n is divisible by 3 for all n E N.

A: We prove that n3+2n is divisible by 3 for all natural numbers n.

Q: (i) Use the method of Mathematical Induction to show that for all positive integer values n21, (1+…

A: Solution of given problem is followed by

Q: (i) Use the method of Mathematical Induction to show that for all positive integer values n21,…

A:

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

A:

Q: For all integers n> 0, 32n – 1 is divisible by 8.

A:

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

A:

Q: Let n be any integer. If n=3k+5 for some integer k. Does 91 6n-3 No, because 6n-3= 18k+30-3 which is…

A:

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

A:

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 <n < 20. Circle any numbers that are…

A: Composite numbers contain prime factors other than 1. Such as if a number has prime factors of 2, 3,…

Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

A:

Q: Prove that 6-1 is divisible by 5 for any n 21.

A:

Q: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: 3. Show that 13" – 7ª is divisible by 6 for n > 1.

A: As per Bartleby guidelines we can solve any one question at a time. Hope you understand.

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

A:

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

A:

Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

A:

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

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

A:

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

A:

Q: C) Prove that the product of k consecutive positive integers is divisible by k!

A:

Q: (a) Prove that 82n – 12" is divisible by 13 for all nɛ N.

A: As per guidelines we are supposed to solve maximum one questions at a time.kindly post another…

Q: By Principle of Mathematical Induction prove 5n – 1 is divisible by 2 for all n >= 1

A:

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

A:

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

A:

Q: Prove by contradiction, with example problem: "For any integer n, if n2 is divisible by 4, then n is…

A:

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

A:

Q: A. Prove that n3 + 2n is divisible by 3 for all integers n.

A:

Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

A:

Q: positive

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

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

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

A:

Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

A: