Prove by induction that 11^n - 6 is divisible by 5 for all n greater than or equal to 1

Q: Show that there are infinitely many primes.

A:

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: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: Show that 8(-n--8()

A: By properties we know that,

Q: Show that the sum of all  odd  integers between 1 and 1000 which are divisible by 3 is 83667

A:

Q: What is the remainder when 2021202120212021 is divided by 2020? use congruence modulo

A: a = 2021202120212021 b = 2020 If a = qb + r then  a ≡ r mod n.

Q: Pigeonhole Principle Prove that among any 100 integers there always two whose difference is…

A: The given problem is to prove the following statement which follows the pigeonhole principle, we…

Q: Rationalize the denomi 20 5/10 - 7/5

A: 20510-75

Q: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

A: Let x be the required number. The number is divisible by 3, so it leaves the remainder 0 when it is…

Q: Prove by mathematical induction

A: The principle of mathematical induction is a technique or method that is used to prove the statement…

Q: Verify using fast exponentiation that the congruence 12964026 = 15179 (mod 64027) is true. Can you…

A: Check the congruence relation       12964026≡15179 mod 64027 using fast exponentiation. Also…

Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: V nEN, 7"- 3n is a multiple of 4.

A: We have to find   For all n= natural number we have to

Q: A number N is divisible by 4 iff the number formed by the last 2 digits of N is divisible by 4. For…

A: We can find number divisible by 4  by using the concept given in the question

Q: Use generating functions to determine the number of differentways 15 identical stuffed animals can…

A:

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: Which positive integers less than 20 are relatively prime to 20?

A:

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

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Q: Derive an expression for the sum of the first n positive even integers.

A: We Use the A.P series. Find The sum of the first n positive even integer.

Q: Explain how to expand (a + b)' using the binomial theorem n an-kbk. Be specific. k expression, k-0

A: Given:      a+b7 The binomial theorem:        a+bN=∑n=0NNnaN-nbn where Nn=N!n!N-n!

Q: Prove through mathematical induction that...

A: Mathematical induction is a method of proving a statement, formula, or expression. In this, the…

Q: What is the 5th term in the expansion of

A: The 5th term of the given expansion is solved by using binomial expansion formula (X+Y)^n.

Q: Suppose 50 cars start at a car race. In how many ways can the top 3 cars finish the race? The number…

A:

Q: Determine the smallest integer that can be expressed as a sum of two primes in three different ways

A: Let us have smallest prime numbers  are, 2,3,5,7,11,13 17,19.....

Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

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

A:

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: Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

A:

Q: Use mathematical induction to prove: For each natural number n, 6 divides (n3-n). I need to write a…

A:

Q: Suppose 59 cars start at a car race. In how many ways can the top 3 cars finish the race? ... The…

A: There is total 59 cars.

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: Prove: The product of 2 numbers is equal to the product of their least common multiple and their…

A:

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

A:

Q: Prove that the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

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

A:

Q: divisible by 3

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: Find the average of the first 50 natural numbers is……………….

A:

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: Determine the smallest integer that can be expressed as a sum of two primes in three different ways.

A: The number is 22 and all others smallest integer can not be written as sum of two primes in three…

Q: (2) Explain why the rational numbers are those having periodic representation or with- out infinite…

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Q: find the sum of all integers between 300 and 800 that are exactly divisible by 14

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Q: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

A: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

Q: Fermat's theorem is true for only prime number. إختر واحداً.

A:

Q: How many natural number that are less or equal to 1000 and that are not multiplea of 3,5or7 exist

A: To find: Number of natural numbers ≤1000 that are not multiple of 3 or 5 or 7.

Q: Without expanding completely, find the indicated term(s) in the expansion of the expression.

A: Given expression: 3c+c247 Concept used for the general term:…

Q: Fermat's theorem is not true for some prime number. اختر واحد

A: False