Is the set O of all odd counting numbers (1, 3, 5… ) closed for multiplication? Why or why not?

Q: "If a and b are positive integers, a divides b and b divides a, then a = b."

A: To Prove: '' If a and b are positive integers, a divides b and b divides a, then a = b''.

Q: Queen chooses sixteen different numbers from the set {1, 2, 3,..., 25, 26}. Prove, or disprove, that…

A: There are total 26 numbers in the set  {1, 2, 3,..., 25, 26}. Also the queen chooses 16 different…

Q: How many numbers of four digits can be formed with the digits 0, 1, 2, 3 (repetition of digits being…

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Q: In how many ways can four distinct numbers be chosen from the set {1,2,...,14} so that no pair among…

A: solution:- Probability when all four chosen numbers are not consecutive = 1 − Probability when all…

Q: 14. Complete. a. (- 3, 2) + (7, – 11) = _?_ b. 3(4, – 1) + (- 2)(-5, 3) =

A: Given a.)(-3,2)+(7,-11)=? b)3(4,-1)+(-2)(-5,3)=? We need to complete the solution.

Q: Prove: The sum of an odd number of odd numbers must be odd.

A: Let us prove this by induction, that if n is odd and a1,a2,. . .. .an are odd, then ∑i=1nai is odd.…

Q: Which represents equal sets?. O A EB and BEA O IAI-IBI O All numbers in Set A are in Set B. O MCNand…

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Q: How may elements are in the set [-2, 2] \N?

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Q: In how many ways can the letters of the word 'RETAIL' be arranged in such a way that the vowels…

A: The objective is to find out in how many possible ways can the letters of the word 'RETAIL' be…

Q: How many natural numbers that are less or equal to 1000 and that are not multiples of 3, 5, or 7…

A: A number is said to be divisible by other number if on dividing the number  with it given remainder…

Q: how many ways can Dr. Yes invite 8 of his friends if he has to invite at most 4 at a time?

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Q: How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, and 9, which are divisible by 5,…

A: Given information- We have given the digits- 2, 3, 5, 6, and 9. We have to form the 3 digits numbers…

Q: Assume that you select digits from the set {1,2,3,4,5,7,8} to form a number between 10 and 800. (1)…

A: given set of digits is  {1,2,3,4,5,7,8}  and the range of numbers is 10 to 800. There would be 2…

Q: Use any method, list all the three digit numbers from the set {3,4,5,6} where there are no repeating…

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Q: How many numbers must be selected from the set {2,4,6,8,10,12,14,16,18,20} to guarantee that at…

A: Use the principle of inclusion and exclusion to solve this problem, as in next step.

Q: (b) How many three digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7 and 8 such…

A: Given Digits: 0,1,2,3,4,5,6,7 & 8

Q: In a group of people, 68 people can speak English, 45 people can speak in Arabic and 19 people can…

A: It is given that 68 people can speak English, 45 people can speak Arabic and 19 people can speak…

Q: Assume that you select digits from the set {2,3,5,7} to form a number between 20 and 700. (1) How…

A: The given set of digits is 2,3,5,7 and the range of numbers is 20 to 700. There would be 2 digit…

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: Show that if seven integers are selected from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, there must be…

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Q: A multiset is similar to a set, but can contain an element more than once. How many multisets can we…

A: You can identify every multiset with an ordered pair of usual sets, call them high and low. If you…

Q: [In(29n)+13no 33n+sin(n) n=1 Consider the sequence Answer the following questions: (i) Does the…

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Q: Prove the statement that "The sum of three consecutive integer numbers is odd if and only if the…

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Q: How many 5- digit numbers can be formed using the digits 0,1,2,3,4,5,6,7,8,9, if repetitions of…

A: Givendigits = 0,1,2,3,4,5,6,7,8,9No.of digits =10

Q: What is the minimum number of students required in a Mathematics class test to be sure that at least…

A: We need to use the pigeonhole principle.  We need to find the minimum number of students that are…

Q: How many ways are there to list the digits 1,1,2,2,3,4,5 so that the digit 3 and 4 are not in…

A: Given Data: To find : How many ways are there to list the digits 1,1,2,2,3,4,5 so that the digit 3…

Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: how many 3-digit number we can form using 10 digits (0,1,2,3.......,8) if no repetition is allowed

A: The first digit have 9 choices (digits 1 -9) as zero cannot be the first digit.There will be 9…

Q: How many 10 digit phone numbers, Area code + number (***)***-****, are possible if the second digit…

A: Total 10 digit number are possible are, the first digit can not be '0' so it has '9'…

Q: Let ß = (1,3,5,7,9,8,6)(2,4,10). What is the smallest positive integern for which ßn = ß-5?

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Q: How many 4-digit numbers greater than 5000 can be formed from the digits 2, 4, 5, 7, 8, and 0?…

A: How many 4-digit numbers greater than 5000 can be formed from the digits 2, 4, 5, 7, 8, and 0?…