3. Show that if m and n are integers and nº + 5 is odd then n is even using a) a proof by contraposition b. a proof by contradiction.

Q: Show that if n is an integer and 3n3 + n is odd then n is odd, by using a) Proof by Contrapositive…

A: To prove i - if n is integer and 3n3+n is odd then n is odd a) By contrapositive :   (P→9)≡(~q)…

Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

Q: Prove by Contrapositive: "For all integers a and b, if a+b is odd then a is odd or b is odd."

A: The method of contraposition is one of the basic methods to a mathematical statement that includes…

Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

A:

Q: Determine whether the following statement is true or false. Justify your answer with a proof or a…

A:

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

A:

Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: 17. Prove that n is square - free every prime number to sat:sfies does not divide ).

A:

Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

A:

Q: Let A be the set of positive integers that are less than or equal to 40 and are multiples of 4 and…

A: Given: Let A be the set of positive integers that are less than or equal to 40 and are multiples of…

Q: How would you prove for any integer n, n(n² - n)is proof by counter example proof by cases since n…

A: We need to select the correct way of proving that for any integer n, n(n2-n) is even.

Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) The 758 ppose

A:

Q: show

A: Given : m+n and n+p are odd  So, we can write, m+n = 2s+1 and, n+p = 2t+1, where, s & t are…

Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

Q: 24. Show that if p is a prime and k is an integer such that 1sksp-1. then p divides ().

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

Q: 2. Prove by contraposition: For all integers n, if (3n +4) is even, then n is even.

A: Contrapositive of 'if p then q' is if not q the not p'

Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

Q: Prove the statement by the method of exhaustion: For each integer n with 1 less than or equal to n…

A:

Q: 1. How many positive integers of 5-digit-number can be formed from the prime numbers starting at 1…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

A:

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

A:

Q: Show that if n is an integer and 3n3 +n is odd then n is odd, by using a) Proof by Contrapositive b)…

A:

Q: Prove that using a proof by contraposition, “if n is an integer and ? 3 + 5 is odd, then n is even”

A:

Q: (4) Consider the statement "There exists an even integer n, such that for all rational number q, we…

A: Statement "There exists an even integer n, such that for all rational number q, we have n²>q"

Q: 3. Show that if m and n are integers and n +5 is odd then n is even using a) a proof by…

A: We have to show that if m and n are integers and n3+5 is odd then n is even using (a). A proof by…

Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &  b/c  , we know there exists integers K1 & K2  ,with K1 a=K2 b=c--1Also…

Q: If p>=q>=5 and p and are both primes, prove that 24| p^2-q^2.

A:

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

A:

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: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

A:

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

A:

Q: Let n be an integer. Prove that if 7 divides (n – 2) or 7 divides (n – 5), then 7 divides (n² + 3).

A:

Q: 4. Let a, b, and d be integers such that dļab. Is it true that either da or d b? If you claim that…

A: Given integers a,b and d. We need to check whether if d/

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: prove by contrapositive : for all n,m integers if nm is even then n is even or m is even

A:

Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural

Q: 1. Prove that if m + n andn +p are even integers, where m, n, and p are integers, then m + p is…

A:

Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: 1- The total number of positive integers that can be formed from the digits 1,2,3 and 4 if no digit…

A: NOTE- AS PER POLICY I  HAVE CALCULATED FIRST MAIN QUESTION ONLY. Here Given 4 digits 1,2,3,4

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: 21. Prove that if n and k are natural numbers such that k < n, then (*:) - (-) - () n k k

A:

Q: Let m, n, p be integers. Use direct proof to show that if m+n and n+p are odd then m+p is even.

A:

Q: 3.3. Proof that for all integers numbers n, if n3 is odd then n is odd, using proof by…

A:

Q: 24. Suppose that I choose six of the first ten positive integers. Prove that I must have chosen two…

A: Rewrite the first ten numbers as follows.

Q: 1. Prove that 3 is irrational number.

A:

Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

A:

Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.