Suppose an integer x is divisible by 21. What other positive integers must also divide x? [Hint: Think about the transitivity of "divides." Enter your answer as a comma-separated list.] Suppose an integer a is divisible by 16. What other positive integers must also divide a? [Enter your answer as a comma-separated list.]

Q: Which of the following rules is used to find the number of positive integers less than 1000 that are…

A: To Determine: which of the following rules is used to find the number of positive integers less than…

Q: proper divisors of $12$ are $1,$ $2,$ $3,$ $4$ and $6$. A proper divisor of an integer $N$ is a…

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Q: 6. Select all the expressions that could be used to find the area of a field that is 1,235 yards…

A: Note:-Since you asked multiple questions, according to our policy we can answer only first question…

Q: Explain why, in any set of 11 positive integers, some two integers must have the same I's digit.

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Q: What is the converse of: an integer has divisors other than 1 and itself, if an integer is…

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Q: Use the given information to find n(A x B) and n(B x A). n(A) = 33 and n(B) = 4 Find the number of…

A: If we have two sets A, B such that  the cardinality of A =m and, the cardinality of B =n then, the…

Q: If x is divisible by 5 or y is divisible by 5, then xy is divisible by 5.

A: Let x is divisible by 5. Then x=5u for some natural number u.

Q: Question 14 Evaluate the following. (DO NOT USE COMMAS WHEN YOU V FOR EXAMPLE ENTER 3456789 NOT 3,…

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Q: Give a non-zero numerical counterexample to the following statement: Every multiple of 12 is also a…

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Q: Suppose you want to refute the statement:  For all natural numbers a, b, and c, if a | b and a | c…

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Q: Prove the statement below directly from the definition of divisibility. For all integers a, b, and…

A: Suppose a|b and a|c. This means that there are numbers f and g such that af=b and ag=c.

Q: Prove or disprove the following statement: the product of any 4 consecutive integers is divisible by…

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Q: Let A = (a, b, c, d) and let # be a binary operation from A x AA presented by the following table: b…

A: Given that

Q: let a, b, and c be integers. if a divides c and b divides c, must a*b divides c?  Prove or come up…

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Q: Prove or disprove: If an integer is divisible by 6 and by 10, then it is divisible by 60.

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Q: Give an example of a set of numbers that is: a. Closed under both addition and multiplication. b.…

A: a. Set of real numbers b. Set of odd natural  numbers c. Set of irrational numbers

Q: 4. Does 3 divide (3k + 1)(3k + 2)(3k + 3)?

A: We will show by induction, put the value of k = 0 we find, (3x0+1)(3x0+2)(3x0+3) = 6  Which is…

Q: Use the following sets: A={Factors of 50} B={odd numbers} Which elements are found in A∩B?

A: A={Factors of 50} = { 1,2,5,10,25,50}  B={odd numbers} ={1,3,5,7,9,11,......}

Q: Write an element from the set B × A × C

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