9. Prove that the greatest common divisor of two positive integers divides their least common multiple.

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: Find all primes that are the difference of the fourth powers of two integers.

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108.

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Q: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

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Q: Find the number of positive integers X smaller than 12849 which have exactly 2022 positive divisors

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Q: How many positive integers less than 850 are multiples of 8, 3, or 5? Use the Principle of…

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Q: A natural number greater than 1 that has only itself and 1 as factors is called a/an number.

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Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

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Q: Consider all positive integers that are mueiples of 20 and that are less than or equal to 300. What…

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Q: 4. Describe all zero-divisors and units of Z6 O ZOQ

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Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

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Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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Q: does there exists   ______such that if n is divided by 5 the remainder is 2 and  if   ?

A: We have to check whether there exist n such that if n is divided by 5 then remainder is 2. Let if n…

Q: Show that if any eight positive integers are chosen, two of them will have the same remainder when…

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Q: Determine the smallest subring of Q that contains 2/3.

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Q: Which positive integers less than 20 are relatively prime to 20?

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Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Prove that there is no triangular number which is one less than a multiple of 11.

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Q: 8. Find Icm(143, 227 9. Prove that the greatest common divisor of two positive integers divides…

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Q: Suppose that Hilbert’s Grand Hotel is fully occupied on the day the hotel expands to a second…

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Q: Discrete

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

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Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

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Q: a) How can you find a linear combination (with integer coefficients) of two integers that equals…

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Q: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: Prove that the product of an even number and an odd number is even.

A: Here, objective is to prove that the product of an even number & an odd number is even

Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

Q: Construct four relatively prime integers a, b, c, d such that no three of them are relatively prime.

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Q: The greatest common divisor of two number 1, b(nteger) are. None of the

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Q: 5. Find the number of positive integers less than 700 that are divisible by 3 but not divisible by…

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Q: Prove: The product of 2 numbers is equal to the product of their least common multiple and their…

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Q: The positive integers that are divisors of 68 are

A: We will use multiple method to find all the divisor of the given number .

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Q: Choose two distinct integers, x and y, between 0 and 9. Derive the number of 9- digit integers that…

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Q: What is the set of O of odd positive integers less than 10?

A: The odd positive integers less than 10 are 1, 3, 5, 7, 9.

Q: 3. There are seven students who always compare their CS test grades. They discovered that either the…

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Q: MK18. Show that in any set of 9 integers chosen from the set {2,3, 4, ... , 21, 22}, there are at…

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Q: Show that every odd integer is the difference between two consecutive perfect squares.

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Q: that the set of negative integers divisible by 3 is countable

A: Set of negetivee integers divisible by 3 isS=-3n, n∈ℕf: ℕ→S   defined asf(n) = -3n,To check…

Q: Exercise 5. Prove that given positive integers a and b, there is a positive integer d such that • d…

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Q: "There is no largest integer."

A: Given: To explain the given statement as follows,

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Q: What 2 nonnegative numbers with a sum of 23 have the largest possible product?

A: Given information: The two non negative numbers with a sum of 23.  Calculation: Consider x and y are…