2. Recall that, when n is a positive integer, n! (read “n factorial") means the product 1 × 2 × 3 × · … x n. Prove that if n > 2, then among the numbers n! +2, n! +3,.,n! + n, none are prime.

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

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

A:

Q: Prove that if n is an integer, then n^2 ≥ n (hint: use cases)

A: Let n be an integer. To prove that n2≥n. Case 1: n=0. Here, 02=0.  That is, 02≥0 Therefore, the…

Q: Prove by mathematical induction that if n is any positive odd integer, then 1 + 3n is divisible by…

A:

Q: Prove that n! > 2" for n > 4.

A:

Q: 2. Recall that a number m is said to be cube free if d | m for d > 1 implies that d = 1.…

A: Consider the provided question, A number 'm' is said to be square free it  d2m and d≥1 ⇒d=1 That is…

Q: Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the…

A: We have to derive the formula 15+25+......+n5=n66+n52+5n412−n212 by using ibn al-Haytham's method…

Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

A:

Q: 8. Prove that if the integer n has r distinct odd prime factors, then 2" | ¢(n).

A:

Q: Prove that n! > 2" for n 2 4.

A:

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

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

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

Q: b) Prove that 4" 10 by induction

A:

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

A:

Q: Number theory.

A:

Q: 2. Give an example to show that the following conjecture is not true: Every positive integer can be…

A: Every positive integer can be written in the form p+a2, where p is either a prime or 1, and a≥0 We…

Q: n. Prove by mathematical induction that 3 + 7 is divisible by 8 for all integers n >0(

A: Answer 15 :-

Q: 2. Prove by mathematical induction the following If n>=4 is an integer, prove by induction that n²…

A: Concept: Mathematical induction: For each and every natural number n, a statement, theorem, or…

Q: a. Show that for every non-negative integers m, n, N = 7.5" has exactly (m + 1)(n+1) different…

A: (a) Given that N=7m·5n To find : Number of positive divisors. Please post the multiple questions…

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: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: イの Prove by mathematical induction that 3" + 7 is divisible by 8 for all integers n > 0

A:

Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

A:

Q: 3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d |…

A: To solve the given questions we will use the following formulas : 1.   We say that two positive…

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

A:

Q: Prove this: -n is a integer; If 6| n then 18| n^2 +3n

A:

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: 1 If n is a positive integer, then (D-a %3D n!

A:

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

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

Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

A:

Q: 9. Ifn is a squarc-frce integer, prove that r(n) = 2" , where r is the number of prime divisors of…

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

A:

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

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

Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

A:

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

A:

Q: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

A:

Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

A:

Q: 4. Prove that given any integer for n, n3 + 2n will be divisible by 3.

A: We have to prove by induction.

Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

Q: 5.) Prove that given any integer for n, n³ + 2n will be divisible by 3.

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

Q: 1. Prove by mathematical induction, for any positive integer n, n³ + 2n is divisible by 3.

A: as per company guidelines we are supposed to answer only one question with maximum three subparts.…

Q: 9. Show that if n is a positive integer then n Σ k² ( x ) = n(n + 1)2²−² k k=0

A: Concept: Mathematical induction: Mathematical induction is a technique for proving that a statement,…

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

A: Given: 1+3+5+...........+2n-1 is n2

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

A:

Q: 5. Use direct proof technique to prove the following theorem: "If n is an odd integer, then n³ + 1…

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

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.