If X is a positive integer divisible by 6, and X <60 What is the greatcst possibe vaive of X 7

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

Q: show that if x is an integer then x2+x-41= 0 produce prime numbers

A: Here show that if x is an integer then  produce prime numbers.

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

Q: If n2 4 is an integer,prove by induction that n<nl.

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Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

Q: Suppose a, b, and c are integers and suppose a | b and a{c, where a}c denotes that c is not…

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Q: Let a be a positive integer. Then Lcm(2a+1,2a+2)= *

A: The given numbers are 2a+1 and 2a+2. Clearly, the given numbers are two consecutive numbers.

Q: Let a and b be integers. Show that if every integer that divides a also divides b, then a | b. (a…

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Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

Q: a

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Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: Show that for any integer n , n2 – 2 is not divisible by 3.

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: Let a and b be integers, not both zero. Thena and b are relatively primeif and only if

A: By standard theorem in number theory which states that:-

Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

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Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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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: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: Use a proof by contrapositive to prove that, if (x + 12)/x is an integer, then x < 13.

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Q: Show that if x is an integer and x3 + 11 is odd, then x is even using a proof by contraposition.

A: We need to prove that if x is an integer and x3+11 is odd , then x is even by contraposition. The…

Q: integer of the form okts is also but not' conversely

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Q: show that if x is an integer then x^2+x-41= 0 produce prime numbers

A: This question is taken from the quadratic equation in which we have to find the roots of the given…

Q: Use indirect proof to show that if n is an integer and n^3 + 5 is odd, then n is even.

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Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: 2. Possible whenn A- Zero b. Negutive integer C- Negative Yational d- Positive integer

A: We know that Laplace transform of

Q: Prove, directly from the definition of divisibility, that for any integer x#0, x|0, 1|x and x|x.

A: Use the following definition of divisibility to prove the required result. Let a and b be any two…

Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Show that for every integer n, f (n) = (2 n3 + 3 n – 10 n + 5 n) is divisible by 7. %3D (Hint: use…

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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

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Q: Use a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

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Q: Suppose that m and n are relatively prime and r is any integer. Showthat there are integers x and y…

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Q: Let n = integer; 7 divides n if and only if 7 dividesn^2 (The result(s) ought to be integer(s))

A: Consider an integer n. We have to prove that 7 divides n if and only if 7 divides n2. Suppose that 7…

Q: show that no integer of the form n^3+1 is a prime other than 2

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Q: 2. Show that if a is an integer other than 0, then a) 1 divides a h) a divides 0

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Q: Suppose a, b, and c are integers and suppose a |b and ałc, where a{c denotes that c is not divisible…

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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: Let be a prime number (integer). Show that p2 + 5 cannot be a prime number.

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Q: show that there does ot exist a rational numbers such that s2 = 6

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Q: Let suppose integers are randomly selected. Prove that the difference of two of them is divisible by…

A: Solution

Q: Use proof by cases to show that if n is an integer then n^2 ≥ n.

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Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: The Pigeonhole Principle:  If there are p+1 or more pigeons, need to distribute among p pigeonholes,…