3. Prove the statement "6 divides n³ - n for all integers n> 0".

Q: 2 4) Prove that n + 5 > 4 n for all integers, n >0.

A:

Q: Show by contraposition that for all integers n^2, if  is odd, then n + 1 is even.

A:

Q: 1. Use a direct proof to show that "If n is an odd number, then n' is an odd number."

A: As per the guidelines, we are supposed to answer only one question at once. Kindly repost the…

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

A:

Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

A:

Q: Find all integers n such that (n^2) - 1 is prime.

A: n² - 1 = (n-1)(n+1)  Now, if (n² - 1) needs to be a prime, then, either n-1 is equal to 1 or, n+1 is…

Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: 9. This question is a little more challenging. How many integers in the interval [1, 99 999] (i.e.,…

A: Given data:   The given sum of all digits is k=10. The number of digits that can be used is n=5.…

Q: b) Prove that 4" 10 by induction

A:

Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: 10. If the product of N consecutive posi- tive integers is divided by N, the remainder is always (A)…

A:

Q: Prove the statement by the method of exhaustion: For each integer n with 1 less than or equal to n…

A:

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 all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

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

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: 16. What is the greatest integer N such that (10 – N) + (16 - N)> 5?

A:

Q: Show that for all integers n:  ceil(n) = -floor(-n)

A:

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

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

A:

Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: 24. Show that if p is a prime and k is an integer such that Isksp-1.then p divides ().

A: See the handwritten solution

Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

A:

Q: Let n be an integer. Prove that if 7 divides (n – 2) or 7 divides (n – 5), then 7 divides (n² + 3).

A:

Q: that for any integer n, n -n is even.

A: Prove that for any integer n, n3-n is even.

Q: Prove that for all integers n, if n² is even then n is even.

A:

Q: Prove that 4 does not divide n2 +2 for any integer n.

A:

Q: Show by contradiction that for any integer n: if n>0, then n+10 n+5

A:

Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # nm.

A: c Given n∈Zand n>2. We have to prove that for all m∈Z, n/m or n+m/nm. Suppose n/m and…

Q: Prove that if 10|n, then 10|n^2+n

A: Prove that if 10|n, then 10|n^2+n

Q: 2. Find a counterexample to the statement n2 – 4 - V nonzero integers n, is not an integer.

A: 2. Given statement is, ∀ nonzero integers n, n2-4n is not an integer

Q: 4. Prove that for every n > 1,3 divides n³ – n.

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: 4. Prove by induction that 21 divides 4"+1 + 52n-1 for all positive integers n.

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

Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

A:

Q: Show that 5 | n5-n for all integers n.

A: To prove the above statement we will use the principle of mathematical induction which is divided in…

Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural

Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

A:

Q: Prove that 30 divides n° – n for every nonnegative integer n. -

A: The given question can be solved as shown in step2.

Q: Show that for all n € N, one has 1 2√n+1 <√n+1-√n<; 2√ñ

A:

Q: Use induction to prove: for any integer n > 1,4 evenly divides 9"

A: for any integer n≥1, 4 evenly divides 9n-1

Q: 4. Suppose that a and b are integers that divide the integer c. If a and b are relatively prime,…

A: Suppose that a and b are integers that divide the integer 'i'If a and b are relatively prime. Show…

Q: 5. Prove or disprove that the following statements are true for all integers n. (a) 2| n? – n.

A: a) prove or disprove:2| n2-n for all integers n.

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

A: