If z ≥ 2 is an integer, show that the following numbers are composite. n! + 2, n! + 3, n! + 4, ... , n! + n

Q: 3. How many 4-digit positive integers are there that are even or contain no O's?

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Q: If n is an integer such that (m² – 1) is odd. Then n is also an odd integer. The following statement…

A: Explanation of the solution is given below....

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: The square of an even integer is congruent to 0 (mod 4). Select one: O True O False

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Suppose that the number of subsets with k elements, made out of a set of n elements is called by the…

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Q: Want a rigorous proof   Prove that The set of prime numbers has no largest element.

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Q: 1. Use a direct proof to show that "If n is an odd number, then n' is an odd number."

A: As per the guidelines, we are supposed to answer only one question at once. Kindly repost the…

Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Find how many positive integers with exactly four decimal digits, that is, positive integers between…

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Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

Q: 9. This question is a little more challenging. How many integers in the interval [1, 99 999] (i.e.,…

A: Given data:   The given sum of all digits is k=10. The number of digits that can be used is n=5.…

Q: Given 10 numbers a1, a2,.… , a10, from the set S natural numbers S = {1,2,3, ., 100} . Prove that…

A: Consider the set S=1, 2, 3, …, 100. Let a1, a2, …, a10 be 10 numbers from the set S. Take the set…

Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

A: Any triple of integers (i, j, k) with 2 ≤ i ≤ j ≤ k ≤ 16 can berepresented as a string of 15 – 1…

Q: If n is an integer and 3n + 2 is even then n is even. (You may use any kind of proof)

A: Given: To prove that if n is an integer and 3n+2 is even then n is even.

Q: i need the last one, thank you

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Q: Show that for all n:  ceil(n) = -floor(-n)

A: In the given question we have to prove that ceil(n)=-floor(-n) That is…

Q: A cosets of of 212 are اخترأحد الخیارات + 2, +1,.1 O + 1,.2 O + 4, + 1, .3 + 4, + 2, + 1 , .4 O

A: Cosets Definition: Let G be a group and let H is a nonempty subset of G. A left coset of H in G is a…

Q: Show that if n is a positive integer then =n(n + 1)2"-2

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: Which positive integer less than 21 are relatively prime to 21? 18 O 19 O 21 O 24 O

A: Which positive interger is less than 21 is relatively prime to 21?

Q: Which list of numbers is ordered from greatest to least? O 3. 49 x 10', 2.73 x 10°, 4. 27 x 10', 1.…

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Q: 2. Use a Venn diagram to illustrate the subset of odd integers in the set of all positive integers…

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Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: Let $S = \{25, 30, 35, \dots, 280, 285, 290\}$. That is, it consists of the multiples of 5 from 25…

A: We have S=25 , 30, ...., 290 Now, the first element in the set is 25 and the last element is 290.…

Q: integers a, b, and c with a≠0, if a∣b,  then a∣(bc). I need a know show table and a proof

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Q: For any positive integers m, n (

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Q: Show by contradiction that for any integer n: if n>0, then n+10 n+5

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Q: Let c = 2^100 and p be the smallest prime such that p >= c. How many composite numbers are between c…

A: Let c=2100 And p=min p is prime:p≥c,p∈ℤ Claim: Total number of composite number between c & p,…

Q: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: All of the following statements about prime numbers are true except

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Q: Find out how many distinct three-digit numbers can be formed using al the digits of 1, 2, and 3. * O…

A: To find  Number of distinct three digit numbers can be formed using all the digits 1,2 and 3.

Q: If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # nm.

A: c Given n∈Zand n>2. We have to prove that for all m∈Z, n/m or n+m/nm. Suppose n/m and…

Q: If n is a perfect number, prove Σ d|n 1 2.

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Q: If a, bị c, d are distinct prime numbers with an as smallest prime the a * b * c *d is Prime number…

A: Given: a, b, c, d are distinct prime numbers and  a is the smallest prime number We know that, the…

Q: Let S be the set of all (nonnegative) 4-digit integers. How many integers does S contain? Enter your…

A: Disclaimer: Since you have asked multiple questions we will solve the first question for you. If you…

Q: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

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Q: How many integers greater than 130 but less than 640 are divisible by 7? O None of these. O 72 O 73…

A: Correct ans is = 73 Please find attachment for the details solution

Q: How many integers between 100 and 999 have distinct digits, and how many of those are even numbers?

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Q: If k, n are postive integers, with k<n, how many subsets of [n] are also not subsets of [k]?

A: Given k and n are positive integers with k&lt;n

Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural

Q: How many positive integers between 100 and 999 inclusive a) are divisible by 7? b) are odd? c)…

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Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: Let p and q be prime numbers. (1) If p is not equal to q, how many generators does Zpq have? (2) If…

A: Given p and q be primes.

Q: In which of the following sets are there in- finitely many prime numbers? {n² – 1 : nɛ \N} {n² – n+…

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Q: Let P be the set of integers that are multiples of 3 between 1 and 10 and Q be the set of positive…

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Q: Let be a prime number (integer). Show that p2 + 5 cannot be a prime number.

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Q: Let S = {25, 30, 35, ., 280, 285, 290}. That is, it consists of the multiples of 5 from .... 25 to…

A: Given that the set S={25,30,35,....,280,285,290} which consists of multiple of 5 from 25 to 290. We…

Q: Prove following statement: For any integer a,the units digit of a^4 is 0,1,5 or 6

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