How many integers from 0 through 60 can be picked such that at least one that is odd? at least one that is even? (a) 31 and 32 (b) 32 and 31 (c) 30 and 31 (d) 31 and 30 Answer A Submit

Q: . Find the sum of all integers between 100 and 500 that are exactly divisible by 12.

A:

Q: (Test for Divisibility by 4) Let n be an integer with decimal repre- sentation a,a … a,a, is…

A:

Q: A prime number p is called special if there exist primes p, P2, P3, P4 such that p = pP + P2 P3 -P4…

A: Prime number problem

Q: Number of integers n with 4 prime factorizations and divisible by 5, where n less than or equal 2021…

A: we shall consider all the integers less than or equal to 2021, and divisible by 5. And also having 4…

Q: 6. A classic unsolved problem in number theory asks if there are infinitely many pairs of "twin…

A:

Q: (a) 1,2, ..., 16, i.e. such that the sum of all numbers in each row, each column and each diagonal…

A:

Q: Consider all positive integers with 3 different digits. (Note that 0 cannot be the first digit.) (1)…

A: Iam assuming that the numbers from 0 to 99 are not valid because they are not 3 digit numbers, so it…

Q: Using direct proof method, prove the following. If n is an even integer then n² + 11 is odd. Answer:

A:

Q: Find the sum of all positive odd integers from 1 to 1239: 1+3+5+7+...+ 1239.

A:

Q: 3. Let n be an integer greater than 2. Explain why n! +2, n! +3, n! +4, n! + (n – 1), n! +n are all…

A: We have to explain that n!+2, n!+3,…,n!+n-1,n!+n are all composite numbers.

Q: wo numbers a and b are chosen from the set 1,2,.., 3n In how many ways can these integers be…

A:

Q: 3. Which of the following integers are divisible by 3? Of those that are, which are divisible by 9?…

A:

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

A: Relative prime are those numbers which does not have any common factor and HCF is 1

Q: Consider the two-digit positive integers 10, 11, 12, ., 99. How many positive two-digit integers are…

A:

Q: Let Sbe the set of all (nonnegative) 4-digit integer. How many integers from start with 1 and end…

A: let S be the set of all 4-digit integer. how many integers from start 1 and end with 9

Q: Fermat's theorem is not true for some prime number. Select one: O True O False

A: Fermat's theorem states that if p is a odd prime number, then for any integer a, the number ap − a…

Q: Prove the statement: For all integers n if 5n is odd, then n is odd. Clearly state the style of…

A: Use method of contradiction.

Q: 3. Formulate and prove a theorem about which numbers can be represented as the sum of 1000…

A:

Q: Using direct proof method, prove the following. If n is an even integer then n²+ 13 is odd.

A: n is even, so take n=2k and put on the expression.

Q: Using direct proof method, prove the following. If n is an odd integer then n+ 13 is even. Answer:

A: Suppose n is an odd integer, so let n=2k+1 for some integer k. Now, we have to prove n2+13 is even.…

Q: 4. Find all the prime numbers that can be expressed in the form n2 – 9 for some integer n.

A:

Q: 5. How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

A:

Q: et S be the set of positive integers less than 500. Find the number of integers in S that are…

A: Given query is to find the number of integers.

Q: Find the sum of the odd integers between 228 and 60

A:

Q: 13. Let S be the set of prime numbers greater than or equal to 2 and less than 100. Multiply all…

A:

Q: che set of integers great

A:

Q: Using direct proof method, prove the following. If n is an even integer then n2 + 13 is odd.

A: Direct proof

Q: You are given 3  ropes and   a lighter.The  time required  to burn through rope 1  with a  single…

A: You are given 3  ropes and   a lighter. The  time required  to burn through rope 1  with a  single…

Q: Proposition. For each integer n, n is even if and only if 4 | n?.

A:

Q: Prove or disprove: let n be an integer. if 3 divides n^2, then 3 divides n.

A: Let us consider some cases for proving that n be an integer. if 3 divides n^2, then 3 divides n.For…

Q: 3. a, =a,,+4", for each integer k21, a, =3 30

A:

Q: Is the set Z of integers countable? Why or why not? How about the set of all odd positive integers?…

A:

Q: Using direct proof method, prove the following. If n is an odd integer then n2 + 13 is even.

A: We have to prove that, If n is an odd integer, then n2+13 is even. Since n is odd, we can consider:…

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

A:

Q: .Prove that if n is an odd number then 3n + 1 is an even number. Use direct proof. 2.Prove that…

A: Given n is odd number

Q: a) Prove that the sum of the squares of two odd integers is always even.

A:

Q: Prove The Pigeonhole Principle(PHP): Theorem. If k is a positive integer and k + 1 or more objects…

A:

Q: sum of all positive integers multiples of 8 between 9 and 99

A:

Q: xercise 4. Let a, b, c be integers. Show that if [a]n = [b]n, then [ac], = [bc],n.

A:

Q: 7. Consider the statement: for all integers a and b, if a is even and b is a multiple of 3, then ab…

A: It is given that a is even . So, a=2m, m∈ℤ Also, b is multiple of 3. So,  b=3n, n∈ℤ ab=2m3nab=6mn…

Q: Using direct proof method, prove the following. If n is an even integer then n2 + 13 is odd. Answer:

A:

Q: Prove or disprove that: (a) The product of an odd number with any integer is odd. (b) The sum of two…

A: We know that there are integers with odd and even values, Odd values include-1,3,5,7,9,11,13,15,17,…

Q: Can 199 cannot be written as the sum of an even integer and two odd integers. Show prove too.

A:

Q: Prove using Direct Proof: For all integers a, b and c if a|b and b|c then a|c.

A:

Q: Find all values of primes p for which (¹¹) = 1 Р

A:

Q: [1] Proof by case: a) Any two consecutive integers have opposite parity. b) Prove: For any integer…

A:

Q: Discrete Mathematics: Prove that every prime number greater than 3 is either one or more or one less…

A: We need to prove that any prime number which is greater than 3 is either 1 more than or 1 less than…

Q: Use proof by cases to show that if n is an integer then n^2 ≥ n.

A: