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

Q: Produce the least positive number that is divisible by​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, and 11.…

A: Given Produce the least positive number that is divisible 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, 11.…

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: How many permutations of the numbers (1, 2, 3, 4, 5, 6, 7, 8, 9} have the numbers 4, 5 and 6 in…

A: So, the numbers 4, 5 & 6 should remain at their natural position and the other numbers can…

Q: 10(c) Suppose a set of 8 numbers are selected from the set {1,2,...,13,14}. Show that two of the…

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Q: ay be formed using the digits 2, 3, 4, 6 and 9 if repetitions

A: 22,23,24,26,29,32,33,34,36,39,42,43,44,46,49,62,63,64,66,69,92,93,94,96,99. In unit digit place we…

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: If repetitions are not permitted, how many 3-digit even numbers can be performed from the six digits…

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Q: How many integers in the set {1,2, ..., 10°} are divisible by 495 and not by 75?

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Q: How many subsets are there of the set N = {5, 42}?

A: We ahave to find no. Of subsets.

Q: let a, b, c, d, four consecutive positive numbers show that g.c.d (a, b, c, d)=1

A: Given :- a, b, c, d, four consecutive positive numbers. As a,b,c,d are consecutive positive numbers…

Q: (b) How many partitions does the set [10]

A: This is a problem ofcounting.

Q: Using the set {1,3,4,6,7,8,9}, how many odd numbers can we construct between 100-999 if no element…

A: An odd number is determined by the unit digit. If the unit digit is an odd number, then the number…

Q: How many different 5-digit odd numbers can be formed from 0, 1, 3, 4, 5, 7, and 9 if repetitions are…

A: Given:Total number of digits = 7 (0, 1, 3, 4, 5, 7, and 9 ) The divisibility rule for 5:The number…

Q: A PIN-number for a credit card consists of a sequence of 4 digits chosen from the set…

A: Given the set {0,1,2,3,4,5,6,7,8,9} We find the 4 digits different PIN number

Q: A. How many 3 digit numbers greater than 200 can be made from 0,1,2,3,4,5 if repetition of digits is…

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Q: 1. How many ways can you choose 4 numbers from the set p = {4, 6, 8, 10, 12} or 3 numbers from the…

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Q: Two distinct numbers are selected from the set {1, 2, 3, 4, . , 36, 37} so that the sum of the…

A: Distinct number: The term distinct number is used to refer to a number in a set that is not equal to…

Q: How many subsets are there of the set N = {6, 49}?

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Q: How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 0, which are divisible by 5…

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Q: (A) For what values of a in (0,8) is f(x) concave down? (If the function is not concave down…

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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?…

Q: Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the…

A: Given digits are 1,2, 3, 4.Given conditions is 1s are seperated by one digit,2s are seperated by two…

Q: How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 8 and 9, which are divisible by 5…

A: From the given Numbers we have to form 3 digit numbers the digits are 2, 3, 5, 6, 7, and 9, which…

Q: How many four-digit odd numbers less than 5000 can be formed using the digits 2, 3, 4, 5, 6. 6, and…

A: Now here given  digits 2,3,4,5,6,9

Q: Consider the list of numbers: 2n-1, where n first equals 2, then 3, 4, 5,6,…What is the smallest…

A: Given that, A numver is, 2n - 1 Where n = 2, 3, 4, 5, 6, 7, ............ We have to find the…

Q: 1) Find the number of positive integers not exceeding 100 that are not divisible by 5 or

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