4. Show that there are no integers m and n such that 80 m + 76 n = 3.

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

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Q: Select two positive integers a and b with 5 <  a  ≤  20  and 5  <  b ≤  20.

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Q: Suppose we define a function f(x) by f(x) = a1 sin(x) + a2 sin(2x) +· + an sin(nx), .. for constants…

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Q: 3. Show that √2+√3 is irrational. Show that √n-1+√n+1 is irrational for every integer n ≥ 1.

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Q: Prove that 5 does not divide 5n+3 for any integer n.

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Q: 16. All positive integers with negative squares are prime.

A: It is a statement. And it is  vacuously true.

Q: determine the value of c(n,5) /p(n,5) where n can be any integer thats 5 or greater

A: Recall the following. Permutations Number of permutations of n things taken r at a time: pn,r=n!n-r!…

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n&gt;1. All prime divisors of n!+1 must be bigger than n. This is…

Q: Disprove  For any integer n, n2 + 5 is not divisible by 5.

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

A: Given, n² is an odd integer. So, n² = 2k+1, where k is an integer. Now, n⁴ = (n²)² = (2k+1)² = 4k² +…

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: 4. Find all the prime numbers that can be expressed in the form n2 – 9 for some integer n.

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Q: 16. What is the greatest integer N such that (10 – N) + (16 - N)> 5?

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Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: What is the smallest integer n for which 25n&gt;512?

A: The provided expression is as follow:

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: For all integers n> 0, 32n – 1 is divisible by 8.

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Q: Show that yn for all n=1, 2₁- ✓

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Q: Show that: 15|(24n − 1) for all natural numbers n.

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Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Determine (with proof) if there exist integers m and x such that 1997m + 47x2 = 3(17x – 1). | If…

A: Given equation is 1997m+47x2=317x-1. To determine: Whether integers m and x exists or not.

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

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

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Q: Prove: “The sum of two even integers is even.”

A: To prove: The sum of two even integers is even. n is an even number if it can be expressed as n =…

Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

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Q: For each pair of integers r and s below, Find the greatest common divisor d. Then, find m, n, such…

A: To determine the greatest common divisor, d and then to find m and n such that d = mr + ns. (a) r=32…

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

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Q: Show that if 9 | a3 + b3 + c3 then 3 divides at least one of the integers a, b, or c.

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Q: Let a, b be positive integers, and let p be a prime integer. Let the notation n|m means m divides n…

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