2. Find all Pythagorcan Triples (not just primitive ones), containing the integer 15.

Q: 1 When two even integers are multipliec, the resulting product is also an ever integer. Answer:Blank…

A: Solution is here

Q: If a,b, and c are consecutive integers and abc=0, then what is the smallest possible sum of there…

A:

Q: Question 3: Prove or disprove: Given integers a, b and c, if a|b and a|c, then a|(b-c).

A: Solution:Given       a | b  and a | cTo prove a| (b-c)

Q: Prove that if n is an integer and n3 + 5 is odd, then n is even

A:

Q: Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for…

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Q: Prove that for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

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: 2- See 3.5 Problem 20: Find all positive integer a>2 such that 2a, 3a +1, 4a+ 2, Sja +3 & 6la +4.

A:

Q: B) Prove that (O) = ) CE) n –k

A: We use the combination formula to simplify both sides

Q: If a, b, and c are integers such that b is a multiple of a³ and c is a multiple of b², then c is a…

A:

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

A:

Q: Find a pair of numbers that provides a counterexample to show that the given statement is false.…

A: We know that, The product of two even counting numbers is an even counting number. The product of an…

Q: Prove that if (a, b, c) is a Pythagorean triple and n is a natural number, then (na, nb, nc) is also…

A: A Pythagorean triple a,b,c has a relation as follows:  c2=a2+b2        .......1

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: For all integers x and y, if 6x+y is odd, then x and y are both odd. True False

A:

Q: Show that if m is an integer then 3 divides m3−m.

A:

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: In Z6, show that 4 | 2; in Z8, show that 3 | 7; in Z15, show that 9 | 12.

A:

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: 1- Show that if a is an integer, then 3 divides a^3-a. with details plz

A:

Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

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

A:

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

A:

Q: Do all four problems. 1. Prove that there exists a unique prime number of the form n2 – 4, where n…

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Q: 2.4 a. Find all the cluster points of the set Q of all rational numbers, and justify your…

A:

Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

A:

Q: Given the set The B= (– 4,7]N ( –1, 10) – {7}· supremum of B is

A: Introduction: In mathematical analysis, the infimum and supremum are concepts that generalize the…

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: If the sum of two counting numbers is an even number, then the product of the two counting numbers…

A: ANSWER:- 1) From the given statement, If the sum of two counting numbers is an even counting number,…

Q: 4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ +…

A: To prove that cube of any integer can be written as the difference of two sqaures.we have given hint…

Q: If a, b, and c are integers such that b is a multiple of a and e is a multiple of b², then c is a…

A:

Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

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

A:

Q: Find out how many distinct three-digit numbers can be formed using al the digits of 1, 2, and 3. * O…

A: To find  Number of distinct three digit numbers can be formed using all the digits 1,2 and 3.

Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

Q: If n is a perfect number, prove Σ d|n 1 2.

A:

Q: 2. Prove that if m and n are integers and mn is even, then m is even or n is even.

A:

Q: Suppose that J and K are points on the number line. If JK = 18 and lies at - 3 , where could K be…

A: NOTE: Refresh your page if you can't see any equations. . the distance between J and K is 18 and the…

Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

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: In your own words, re-create the proof of one of the examples: (prove that √2  is irrational)

A:

Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

Q: Prove in any Pythagorean triple (a, b, c) that either a or b is divisible by 3 and one of a, b, c is…

A: A Pythagorean triple is three numbers a,b,c such that a2+b2=c2. Also if a,b,c is a Pythagorean…

Q: Prove that if m and n are integers and mn is even, then m is even or n is even.

A:

Q: Suppose that and K are points on the number line. IF JK = 13 and lies at 5, where could K be…

A: Recall: Distance cannot be negative but if you will  get negative that means you will move the…

Q: 2 There are integers m and n such that m² +n² is a perfect square.

A:

Q: What is the smallest positive integer x in the equarion: 1=3x+2(mod 7)?

A: We have to solve the equation 1 = 3x + 2 (mod 7)⇒1 - 2 - 3x = 0 (mod 7)⇒-1 - 3x = 0 (mod 7)⇒3x = - 1…