How can I prove by induction that 2^n + 1 is divisible by three for all odd integers

Q: Show that there are infinitely many primes.

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Q: are there any even numbers greater than 2 which are prime? why or why not?

A: We will find the solution in the following step

Q: How many positive integers between 500 and 999 inclusive are not divisible by 5?

A: For this problem, we must recall that to calculate the number of integers in a range of values that…

Q: How many 3 or 4 digit even numbers greater than 700 can be formed by: a. Using the digits 0-9.…

A: EVEN NUMBERS : Even numbers embrace people who conclude in zero,2,4,6, and 8. Even numbers, as an…

Q: Compute the sum of the first 60 positive integers that are exactly divisible by 4.

A: The numbers divisible by 4 is:

Q: A sequence of positive integers starts with one 1, followed by two 2s, three 3s, and so on. (Each…

A: Given:     The sequence is:     1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,... To get this number, we have to…

Q: How many four digit positive integers are there that contain the digit 3 and are divisible by 5?

A: A number is divisible by 5, if the unit digit place contains the number 0 or 5. Let m choices to be…

Q: Let ab5c be all four-digit multiples of number 36 with different digits. If someone found 7 numbers…

A: Let ab5c be all four digit multiples number 36 with different digits.   As 36= 4x9, so the number…

Q: How many positive integers less than 1,000,000 have a digit sum of 5?

A: We have to find the number of positive integers 1,000,000 have a digit sum of 5. The number less…

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: How many 5-digit even numbers can be formed using the digits 2, 4, 6, 7, 9 if no repetition is…

A: Given:

Q: Prove that 2^10 + 5^12 is a composite number.

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Q: All composite number can represent as product of primes numbers

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Q: Compute the sum of all integers between 60 and 150 that are exactly divisible by 8.

A: Consider the given range of number 60 to 150

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: How many pairs of two distinct integers chosen from the set {1, 2, 3,..., 101} have a sum that is…

A: Since you have posted multiple questions and we can answer one question. So answer provided for the…

Q: suppose repetitions are not allowed. Find the number n of three-digit even nimbers that can be…

A: Given problem Given that   Suppose repetitions are not allowed. Given six digits : 2,3,5,6,7,9…

Q: How many positive integers less than 1000 (1) are divisible by 7? (2) are divisible by 7 but not by…

A: # As per the guidelines we are entitled to solve three subparts at a time, please resubmit the other…

Q: How many integers in {1,2, 3, ..., 99, 100} are not divisible by 7? (Give the exact number)

A: The given set of numbers is, 1,2,3,.....,99,100 The total no.of numbers from 1 to 100 is, 100 The…

Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

Q: Show that n-lis divisible by 8, if n is an odd positive integer

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Q: 5. Give a recursive definition of the set of positive integers that are multiples of 5.

A: Solution

Q: Compute the sum of the first 50 positive integers that are exactly divisible by 5.

A: To determine sum of the first 50 positive integers that are exactly divisible by 5.

Q: How many combinations are there to pick r objects from 2n objects numbered from 1 to 2n with n > 2…

A:

Q: Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it…

A: (a)The fill up for the statement to rewrite the given statement: Is there an integer n such that n…

Q: How many integers in the set {1,2,3,4, ..., 200} is divisible by at least one of the primes 2, 5 and…

A: Just use inclusion-exclusion principle for three sets.

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: find the sum of all even intergers less than 1000 that are exactly divisible by 7

A:

Q: 3. Explain the relationship among the three concepts: well ordered principle, induction, strong…

A: We know. Well ordered principle:  A set of numbers is known as well ordered when each of its…

Q: How many pairs of two distinct integers chosen from the set {1, 2, 3, 101} have a sum that is even?

A:

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 positive odd integers less than 10,000 can be written using the digits 3, 4, 6, 8, and 0 if…

A:

Q: How many positive integers not exceeding 3000 are divisible by exactly two of 4, 10, 12? (Steps…

A:

Q: The sum of all positive integers less than 400 which are divisible by 5.

A: Positive integers less than 400 which are divisible by 5 are⇒n=79 5,10,15,20,...,395 The above…

Q: How many integers between 1 and 10000(inclusive) are divisible by at one of 9, 10, 11?

A:

Q: Suppose 26 cars start at a car race. In how many ways can the top 3 cars finish the race? ..... The…

A:

Q: Give a recursive definition for the set of all the positive integers that are less than 100 and not…

A: Given: Set of all positive integers that are less than 100. To find: Recursive definition for the…

Q: How many different words can be formed with the letters of the word ‘SUPER’ such that the vowels…

A:

Q: How many 3-digit numbers with digits in {1,..., 9} (repeats allowed) are divisible by 9?

A: Given digits are 1,2,3,4,5,6,7,8,9. Make three digit numbers using the given digits. To find: The…

Q: How many three-digit odd integers can be formed using the digits 0, 1, 4, 5, 6, 7, and 9 only once?

A: Permutation: A permutation of a set is defined as an arrangement of its members into a sequence or…

Q: Can you think of the smallest number N such that ot can be expressed as a product of the two factors…

A: We have to find out the number N that can be expressed as a product of the two factors which are…

Q: Prove that no integer congruent to 3 modulo 4 can be written as the sum of two squares.

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Q: How many can 3 digits be formed using the digits from 1 to 5 if the digit 2 is never there in the…

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Q: How many 5 digit positive integers are divisible by 5 and have at least one 6 as their digit?

A: A number is divisible by 5, if it has a 0 or 5 in the unit digit place. Let m choices to be made…

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Find the sum of all  natural numbers between 250 and 1000 which are exactly divisible

A:

Q: A natural number greater than 1 that is divisible by a number other than itself and 1 is called a/an…

A:

Q: Let ab5c be all four-digit multiples of number 36 with different digits. If someone found 7 numbers…

A: Let ab5c be all four digit multiples number 36 with different digits. As 36= 4x9, so the number ab5c…

Q: What 2 nonnegative numbers with a sum of 23 have the largest possible product?

A: Given information: The two non negative numbers with a sum of 23.  Calculation: Consider x and y are…