Let P(n) be the assertion that: for any integer n 2 1, 7" - 1 is divisible by 6. Part 1 Part 2. In the inductive step, we assume that for any k> 7Ak-1 7^0-1 is divisible by 6. We will prove that is divisible by 6. Part 3

Q: 24. Which is the recursive definition of the set of positive integers that are multiples of 4? * aj…

A: We have to find the correct option which represents the set of positive integers that are multiples…

Q: 17. Expand using Pascal's Triangle or the Binomial Theorem. Be sure to show the setup. Then,…

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Q: How to write 0.014 (014 recurring) in the form p/q.  (Where p,q are elements of real numbers.)

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Q: [1+6n A" 7 4 4n -9 -5 -9n 1-6n | If , show that for every n natural number.

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Q: Prove the following: (a) If a (b) If a b(modn) and c€ Z then cacb(modn). b(modn) and e=d(modn) then…

A: As per the guidelines I am answering only one question with all its parts.Remember :

Q: Any whole number is divisible by 3 if and only if the sum of its digits is divisible by 3 TRUE FALSE

A: We have to state that the given statement is true or false.   Any whole number is divisible by 3 if…

Q: not use ai please

A: Explanation:1. Then, 3n + 4 = 3(2k + 1) + 4 = 6k + 7 = 2(3k + 3) + 1 (3)2. By definition of odd,…

Q: (a) 1,2, ..., 16, i.e. such that the sum of all numbers in each row, each column and each diagonal…

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Q: For a natural number n, we define n! = 1-2-3 (n − 1). n. - For example: 2! = 1-2, 3! = 1-2-3 = 6,…

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Q: A number N is divisible by 4 iff the number formed by the last 2 digits of N is divisible by 4. For…

A: We can find number divisible by 4  by using the concept given in the question

Q: prove

A: Here we use, 3+3=9 and 6+3=9 We have to prove,3×3=9 by peano's axiom for natural number.

Q: 6. Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 24. Give a recursive definition of a) the set of odd positive integers. b) the set of positive…

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Q: How many three-digit numbers are possible under each condition? (a) The leading digit cannot be zero…

A: We know that 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are digits in the number system.

Q: 5. Let p = 61. Find all incongruent elements modulo 61, that have order: (a) 5 (b) 10 (c) 60

A: p=61 incongruent modulo 61={x:1≤x≤n with gcd(x,n)=1}={1,2,3,4,5,6,7,8,9,10,11,12,13,...60}

Q: p Pq C N-n N-1

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Q: Assume that you select digits from the set {2,3,5,7} to form a number between 20 and 700. (1) How…

A: The given set of digits is 2,3,5,7 and the range of numbers is 20 to 700. There would be 2 digit…

Q: 6) Let n be a positive integer such that 7 | (52n-1 +22n-1). Prove that 7 || (52n+1+22n+¹). (Hint:…

A: Since, given that  52n-1+ 22n-1  is divisible by 7 .  We have to prove that  52n+1+22n+1 is also…

Q: 15, 9,3, -3, how do you find the next term

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Q: rs, and if c a and c| b, then c| (6

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Q: Calculate the right hand side of the statement below for n=12 ©George Mason University 3+4+5+. + (n+…

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Q: find the 100th term of 1, 3, 9,...

A: Our aim is to find the 100th term of 1 , 3 , 9,......

Q: Prove that if n is an integer and 5n + 6 is even, then n is even.

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Q: 5. Consider the subtraction problem "6 - "13. a. Show how to represent the problem on a number line…

A: 6a)

Q: Prove this statement Discrete Math There exists a prime number l so that l^2 − l + 11 is also…

A: We have to prove that, "there exists a prime number I so that I2-I+11 is prime". We know the set of…

Q: how do you proof part a with a formal proof

A: Given a standard deck of 52 cards consists of 4 suites (hearts, diamonds, spades and clubs) each…

Q: Show that given any set of 5 natural numbers, there are 3 numbers whose sum is divisible by 3.

A: Let q∈N. Note that, for every q there exists some x∈N and some v∈0,1,2 such that q=3x+v. Now, let…

Q: Question: Evaluate each following expressions a.) If x =0   evaluate 4x**2 + 5(9 +x) + x **5 b.)…

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Q: Using direct proof method, prove the following. If n is an even integer then n2 + 13 is odd.

A: Direct proof

Q: • Show that if m divides 3, then m divides 9. (We say a divides b if there exists an integer k such…

A: Let  m divides 3 .  Then there exists an integer k such that 3 = mk. Now, 9 can be written as…

Q: Find a formula for the general term a, {5, 8, 11, 14, 17, .} 5n + 3 On+3 O 3n + 2 O 2n + 3

A: We can answer the question as below

Q: If n is a positive integer, then 2 n - [a] »] + [a + ²] + [a + ² ] -- - + [a + ¹ = ¹] - =] = n n =…

A: Given: nis a positive integer. To Prove: α+α+1n+α+2n . . . +α+n-1n=nα

Q: 5 =5 Hence, 6"-1 is divisible by 5 is (10.). for n=1 Part 2: Assume that 6" - 1 is divisible by 5…

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Q: m = q192 ··•¶s – P192··· ¶s %3D .. =n – pi(q2 .qs) - = P1P2•• · Pr – P192 ··•92 .. = P1(P2 .· Pr –…

A:

Q: 2(5) – 3(1 – 6) Is even integer?

A: If an integer is divisible by 2, then it is called an even integer. For example, 2,4,6 etc. are even…

Q: True or False. Briefly explain why.  {−13, −6, −3, 1, 10} is a complete residue system modulo 5.

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Q: USING MODULAR ARITHMETIC PERFORM THE FOLLOWING OPERATIONS    A).  (27 + 7) (mod 4)…

A: "Since you have posted a question with multisubparts, we will solve the first three subparts for…

Q: Theorem 4.6 (Fundamental Theorem of Arithmetic) Every positive integer n can be written as the…

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Q: Using c++

A: Correcting the code:To fix the code, we need to replace the incorrect condition with a logical…