How many positive integers less than 1000 (1) are divisible by 7? (2) are divisible by 7 but not by 11? (3) are divisible by both 7 and 11? (4) are divisible by either 7 or 11?

Q: How many integers between 100 and 400 contain the digot 2?

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Q: Given that the expression (n²+n+1) n! can be written as a(n + 2)! + b(n + 1)! + cn! where a, b, c…

A: The main objective is to find the sum value.

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: How many integers from 1 to 700 (inclusive) are divisible by the factors of 70 greater than 1?

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Q: Use the definitions  to show that the following statements are true. For example, to show that 5 |…

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Q: Find a formula for 1/2+1/4+1/8+.+ 1/2h by examining the values of this expression for small values…

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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: Think about the three-digit numbers that can be formed from the digits 1, 2, 3, 4 and 5 without…

A: Solution

Q: 19. What is the value of (pk) when p is prime and k is a positive integer?

A: We know that the value of the Euler ϕ-function at the positive integer n is defined to be the number…

Q: The set T:{3,5,9, 12) Hret Unde malkplicaion group ti) find the idenhily IV) Establis h nodu lo 15…

A: See the attachment

Q: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

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Q: 5. (12 points) Find how many positive integers between 1 and 99, inclusive, have the following…

A: The objective of this question is to find the number of positive integers between 1 and 99 that are…

Q: divisible by 3 for

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Q: (8) How many integers in the set [1,000] = {1, 2, ..., 1,000} are (a) divisible by 5? divisible by…

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Q: Show that (a) 8n −3n is divisible by 5 for all natural numbers n.  (b) n2 −n +41 is odd for all…

A: expansion of (a+b)n = hco an bo + nc1 an-1 b1 +.......... +ncn ao b n

Q: Disprove  For any integer n, n2 + 5 is not divisible by 5.

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Q: 3. In this problem we use the notation for the digits a; of an integer m given by m = (akak-1...ao)…

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Q: How many numbers between 1 and 10000 have the digits 1,3, and 5 such that each digit appears exactly…

A: Here Numbers between 1 and 10,000 have digits 1,3,5 and each appears exactly once only

Q: (3) Prove that if a cube of a number is divisible by 17, then the number is divisible by 17.

A: We have to prove that if cube of a any number is divisible by 17 then the number itself is divisible…

Q: Prove a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

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Q: 9. 10. What is t divisors?

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Q: True or false: If a is any integer, then (5a + 2,7a + 3) =1. Justify your answer.

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Q: (a) Determine if the following numbers are prime: 101! + 78, 88! + 55 (b) Is 299 + 1 divisible by…

A: A prime number cannot be divided by any other number except 1 and itself. Now use this definition to…

Q: If k is a positive integer, what is the remainder when (k+ 2)(k3 - k) is divided by 6? Stilo…

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Q: We know: ½ of all integers are divisible by 3 and t of integers are divisible by 7. What fraction of…

A: We need to find fraction that will be divisible by 3,7 or both.

Q: True/False: Every integer greater than 1 is divisible by a prime number. True False

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Q: Prove (n + 6)² – (n + 2)² is always a multiple of 8. (Use direct proof.)

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Q: *********** Use the definitions of even, odd, prime, and compositive numbers to justify your answers…

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