For every positive integer a, find a composite number n such that n|a^n - a.

Q: Use mathematical Inductlon to prove that for r>0 and any positive Integer

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Q: Show that the greater than or equal re lation >) iš a partially order on the Set of integer.

A: Partially Ordered Relation: A relation R on set P is said to be partially ordered if and only if R…

Q: Find the least integer n such that (2x^3+x^2logx) is O(x^n).

A: The given expression is 2x3+x2logx. Concept used: Let f and g from the set of integers or the set of…

Q: Find a positive integer n for which the equality holds = 1 2

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Q: Find the term with no x in the expansion of x 15

A: we have to find the term with no x in the expansion of x3+12x-215.  as we know that the (r+1)th term…

Q: Give a combinatorial interpretation of the coefficientof x6 in the expansion (1 + x + x2 + x3 +⋯)3.…

A: According to the given information, it is required to give a combinatorial interpretation of the…

Q: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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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: Prove the formula for the sum of the first positive integers:

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Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Find the Term that free from x of the expansion (x2 +10

A: Given expansion(x2+(2/x3))10 a=x2 and b=2/x3, n=10 Formula  General term formula is, Tr+1= nCr…

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

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Q: Find the number of ways a 5-digit positive integer can be made such that in each of them every digit…

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Q: Prove that 2" >n For all nuatural numbers

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Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: () is even. 3. Prove that for all positive integers n, the number

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Q: Use generating functions to determine the number of differentways 15 identical stuffed animals can…

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Q: Discrete Math

A: According to the given information, Suppose that:

Q: Find the product of all positive integers from n down through 1 for n = 5.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Find all the odd positive integers n which satisfy ó(n) = 20. %3D

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Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: Define an odd number and show that the product of odd even numbers is also even.

A: We need to first define an odd number. And layer prove product of an odd and an even number is…

Q: Find all integer numbers a for which

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Q: 3. Prove that 2" > n For all nuatural numbers

A: Solution

Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: n + 2020 n + 2021 and LVn +1]

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Q: Use generating functions to find the number of ways toselect 14 balls from a jar containing 100 red…

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Q: Find the least positive integer n for which 2" + 5" - n is a multiple of 1000.

A: To find the value of n substitute n=1,2,3... multiple of 1000 is a certain number divided by 1000…

Q: Write “the set of all natural numbers N less than 9” intwo different ways.

A: To write “the set of all natural numbers N less than 9” in two different ways.

Q: Let n be a positive integer. Prove that :)- (:)- +2. +...+n. = n2"-1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

A: Given an=1000nn!. We have to find the N∈I such that an+1&lt;an, n&gt;N. For n=1 a1=100011! For n=2…

Q: Develop a simple expression for the entries of A", where n is a positive integer and 1 A = -IN -IN

A: To get a simple expression for the entries of An, where n is a positive integer. A=112012

Q: Consider all numbers between 20000 and 70000. Efficiently determine: A) How many of these numbers…

A: Part A - Let’s represent the number as ABCDE. You have to considere the special cases first, since…

Q: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: Find the least natural number a with the property that the sum a + 2a + 3a + .. + 9a is the number…

A: Given sum is  a+2a+3a+4a+5a+6a+7a+8a+9a

Q: provide a recursive definition for the set Y of all negative multiples of 7.

A: Consider the provided question, Recursive definition: Recursion is the general definition term for…

Q: Find integers x and y such that gcd(N, 11353) = Nx + 11353y.  Let N = 116023

A: The greatest common divisor(gcd) of a prime number and any other number is always equal to 1.

Q: Use a direct proof to show that the product of two odd numbers is odd.

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Q: Find odd and general

A: Given,       f(x)=x-π3;   0≤x≤π

Q: be run under the building. If these two blocks must always be next to each other, how many s can the…

A: Given Here we have 7 blocks ,but we  have to keep 2 perticular blocks together. So we have to…

Q: Show that |x^n| = |x| ^n for every positive integer n and every real number x.

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Q: Find two positive integers such that the four times loo0 and the product large as possible. the .…

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Q: Define β1 stands for?

A: β1 is mainly used in linear regression.  The linear regression is defined as the mathematical linear…

Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: Find mZD. 6. D Write your answer as an integer or as a decima mZD = %3D

A: We have a right-angled triangle right angled at F we have to find angle D

Q: For n > 3, prove that the numbers ±5, ±52, · reduced residue system modulo 2". form a

A: A subset R of the integers is called reduced a residue system modulo n if (i) gcdr,n =1 for each r…

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…