prove by induction 12. For any integer n >= 0, 7^n-2^n is divisible by 5.

Q: Show that there are infinitely many primes.

A:

Q: are there any even numbers greater than 2 which are prime? why or why not?

A: We will find the solution in the following step

Q: show that SZ+5 is irrational number

A: If a number cannot be written in ab form, where a and b are integers is known as an irrational…

Q: There are 8 people in a room, whose ages are between 10 and 40. Show that we can always find two…

A: Pigeonhole principle Here, we don't know the ages of 8 people, but we know the range of sum of their…

Q: Distinguish between finite and infinite?

A: Finite means that the quantity is limited . It does not mean that it can be calculated on the piece…

Q: Without using any theorems from the book, prove the following statement: The sum of any two odd…

A:

Q: Suppose the repeating in 0.020020200202002.... continues forever. Is this number rational?

A: We can not write the number 0.020020200202002... in the form  p/q where p is an integer and q is a…

Q: Find all three-digit numbers with different digits with the property that those numbers are…

A: Given that it is to be found all three-digit numbers with different digits with the property that…

Q: Prove by mathematical induction

A: The principle of mathematical induction is a technique or method that is used to prove the statement…

Q: How many positive integers less than 850 are multiples of 8, 3, or 5? Use the Principle of…

A: Using the Principle of inclusion/Exclusion, the formula for A∪B∪C can be written as:…

Q: When four positive integers are added three at a time the sums are 1551, 1659, 1720 and 1736. The…

A: when four positive integers are added three at a time the sums are 1551, 1659 ,1720 and 1736. we…

Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: A sequence of positive integers starts with one 1, followed by two 2s, three 3s, and so on. (Each…

A: Given:     The sequence is:     1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,... To get this number, we have to…

Q: V nEN, 7"- 3n is a multiple of 4.

A: We have to find   For all n= natural number we have to

Q: Suppose A = (a,) = (a1, a2,as....) is an increasing sequence of positive integers. A number e is…

A: A = an = 2n =  ( 1,2,4,8,16 . . .  ) is an increasing sequence of positive integer. 1 = 1  2 = 2 3 =…

Q: Give a combinatorial interpretation of the coefficientof x4 in the expansion (1 + x + x2 + x3 +⋯)3.…

A:

Q: By using the principle of mathematical induction, prove that 1² + 2² + 3² + ..... + n² = fraction…

A: In this question, we prove the given statement  12+22+.....   + n2=n(n+1)(2n+1)6 By using…

Q: How many 5-digit even numbers can be formed using the digits 2, 4, 6, 7, 9 if no repetition is…

A: Given:

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

Q: All composite number can represent as product of primes numbers

A:

Q: Let n and k be positive integers with 2k + 1 ≤ n. Find the number of ways to choose k children from…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

Q: Compute the sum of all integers between 60 and 150 that are exactly divisible by 8.

A: Consider the given range of number 60 to 150

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

A:

Q: Find: The sum of all positive integers less than 400 which are not divisible by 5.

A:

Q: What is the 5th term in the expansion of

A: The 5th term of the given expansion is solved by using binomial expansion formula (X+Y)^n.

Q: Determine the smallest integer that can be expressed as a sum of two primes in three different ways

A: Let us have smallest prime numbers  are, 2,3,5,7,11,13 17,19.....

Q: how to prove √6 is irrational

A:

Q: prove that "The Sum of two odd numbers is even: using direct. Note:- 1 want the ansawer on a paper.

A: The sum of two odd numbers is even. Proof :          Properties odd and even integer.          If…

Q: Show that, selecting 91 positive integers guarantees that there are at least 10 integers with the…

A:

Q: Determine the probability that the randomly selected number from the interval [0,1] is a rational…

A: To Determine: The probability that the randomly selected number from the interval [0,1] is a…

Q: Use mathematical induction to prove: For each natural number n, 6 divides (n3-n). I need to write a…

A:

Q: Sort this permutation [5 213 4] into identity permutation [123 4 5]. Check what's the number number…

A: The permutation is nothing but the arrangement of objects in different orders. If we permute a set…

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

A: (a)The fill up for the statement to rewrite the given statement: Is there an integer n such that n…

Q: Use permutations to solve. From a cooler with 9 different kinds of soda, 3 are selected for 3…

A: The formula of permutation is: Prn=n!n-r! This is the required answer.

Q: How many positive integers not exceeding 3000 are divisible by exactly two of 4, 10, 12? (Steps…

A:

Q: In two or more complete sentences, explain how to find the third term of the expansion of (2x+y)4.

A:

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: Find the average of the first 50 natural numbers is……………….

A:

Q: Determine the smallest integer that can be expressed as a sum of two primes in three different ways.

A: The number is 22 and all others smallest integer can not be written as sum of two primes in three…

Q: Find odd and general

A: Given,       f(x)=x-π3;   0≤x≤π

Q: (2) Explain why the rational numbers are those having periodic representation or with- out infinite…

A:

Q: Fermat's theorem is true for only prime number. إختر واحداً.

A:

Q: 2- Using Direct proofsprove if odd intege mis an even integer then mn is also an even integer

A:

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Without expanding completely, find the indicated term(s) in the expansion of the expression.

A: Given expression: 3c+c247 Concept used for the general term:…

Q: Show that 19 divides 10*8n-1 + 33n-1 for every positive integer n.

A: Concept: Mathematical induction: For each and every natural number n, mathematical induction is a…

Q: Let ab5c be all four-digit multiples of number 36 with different digits. If someone found 7 numbers…

A: Let ab5c be all four digit multiples number 36 with different digits. As 36= 4x9, so the number ab5c…

Q: Fermat's theorem is not true for some prime number. اختر واحد

A: False

Q: algorithm to determine the decimal number answer to 2893 ÷ 6 to the hundredths place. b. Interpret…

A: Solution of the problem as follows