1. (20 pts) Given n is a positive integer, prove that n is odd if and only if n² + 3 is even. You must use definitions of even (and odd, if needed). Do not use "even plus even is even" or similar rules of thumb.

Q: An operation * is defined on the set of real numbers by a * b = a + b – ½ ab. What is (3 * 2) * 2? *

A: The rational number and irrational number are together called real numbers. An operation is a rule…

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Q: Which of the values below is closest to 3! +4! + 5!? O 100 O 150 O 200

A: Given  which of the values below is closest to 3 !+4!+5!

Q: Evaluate. n=1 2n+4 7n

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Q: Fill in each blank so that the resulting statement is true.

A: We have, 3x=π4+nπ  for any integer n.

Q: the statement below true or false for n=1? ©George Mason University 2+ 4 + 6+ ... + 2n = n² + n + 1

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Q: "If a and b are positive integers, a divides b and b divides a, then a = b."

A: To Prove: '' If a and b are positive integers, a divides b and b divides a, then a = b''.

Q: 19. What is the value of (pk) when p is prime and k is a positive integer?

A: We know that the value of the Euler ϕ-function at the positive integer n is defined to be the number…

Q: Using the result you proved in the question above, find a list of 200 consecutive composite…

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Q: How many different numbers which contain from thr an be formed from the digits A, B, C, D and E. and…

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Q: 4 (ii) 16 | at + 15.

A: Step-1 1)16|a4+15

Q: . Homework. Let a and b be integers. If 7 | (17a +256), prove that 7 | (10a+11b).

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Q: Find integers ? and ? such that 3220? + 6006? = gcd(3220,6006).

A: Remark: The greatest common divisor (gcd) of two integers is the greatest integer that divides both…

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A: Solution

Q: 1. Say whether the following is true or false and support your answer by a proof. (3m N) (n = N) (3m…

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Q: the root lies between 2.625 and 2.75 is 2.6875. O True O False

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Q: 5+2-5+p 13) 3. 3) 28

A: Given: The equation 53p+23=5+p2p, To determine: The value of p,

Q: 4. Proofs. Please write clearly. If I can't read your handwriting I will not be able to follow the…

A: The problem is belongs to the classical algebra. we have to prove the first problem by contradiction…

Q: 3S+8 2) L S42845 s428+5)

A: We need to find , Inverse Laplace Transform, of F(s) = 3s + 8s2 + 2s + 5 Since , F(s) = 3s + 8s2 +…

Q: B be a real number such that B+ is an integer. Show that Br + is an integer all natural numbers n.

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Q: se indirect proof to show that, "If n is an integer and 3n + 2 is even, then n is even

A: Let us solve the classical algebra problem in the next step.

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n n + 1 %3D k +1, j=k

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Q: How many integers in between 1 and 3000, inclusive are divisible by either 5 or 7?

A: Let A and B are two sets such that then number of elements in A and B are obtained as…

Q: : a number is even  if

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Q: The value of n such that ph pn- ,n> 4 is 15.

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Q: Q2 1-The Euler's number (e) is a * rational number a. True b. False

A: Rational numbers are those which can be represented in the from of fraction and those which can not…

Q: Find the value of n(A U B) if n(A) = 8, n(B) = 13 and n(A n B) = 6. n(AUB) (Type a whole number.) =

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