(a) Determine if the following numbers are prime: 101! + 78, 88! + 55 (b) Is 299 + 1 divisible by 27? 5.

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A: Given number is : 613 To find what is the minimum amount of number must divide into 613 to check…

Q: Let p>5 be the prime number. Prove that the expression p4-10p2+9 is divisible by 1920.

A: Let, p>5 be the prime number. Prove that the expression p4-10p2+9 is divisible by 1920.

Q: 15) n(n*+5) is divisible by for each integer

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A: Prime numbers are the numbers which have only two factors one factor is 1 and other factor is 2

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A: The number N is calculated by the prime…

Q: If mis odd, then m3 - mis divisible by 3.

A: We will check this for m=1 first  0 is divisible  by 3.

Q: 2. 6n - 1 is divisible by 5 for n any positive integer

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Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

Q: The number of factors of prime numbers are Depends on the prime number None of the mentioned (3 2

A: Given query to find number of factors of a prime number.

Q: Let p > 5 be the prime number. Prove that the expression p – 10p +9 is divisible by 1920.

A: We have to prove that the expression p4-10p2+9 is divisible by 1920.

Q: Let be the prime number p>5. prove that the expression p^4-10p^2+9 is divisible by 1920

A: First we will expand the given expression p4-10p2+9⇒ p4-10p2+9=p4-9p2-p2+9⇒…

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

A:

Q: The number of factors of prime numberss are (None of the mentioned Depends on the prime number (2 (3

A: Prime number is only divisible by 1 and that number itself

Q: Find the Prime Factorization of 648, 1056, and 1496. Provide at least one divisibility test for each…

A:

Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: 2) 10! + 1 is not divisible by 11 a) True b) False a) True O b) False O O

A: A number is said to be divisible by another number if we can divide the number without getting any…

Q: Show that for any integer n , n2 – 2 is not divisible by 3.

A:

Q: Show by induction that 19"-8" is divisible by 11 for all n >0

A:

Q: composite. 10. Suppose that n = 2³ × 5² × 7 × 17³. Give the prime factorization of n² and nº (Hint:…

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Q: Prore the only prime of the form n°-4 is 5.

A:

Q: Is the conjecture "all prime numbers are odd" true or false? O True O False

A:

Q: b) If x is an even prime number, then x is divisible by 2. Converse: Inverse: Contrapositive:

A: Given the statement, "If x is an even prime number, then x is divisible by 2". It is of the form,…

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

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Q: If n is prime, then 2" – 1 is prime.

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Q: The Great est Common Divisor of two prime numbers a and bis a + b ab a b O 1 0

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

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Q: Find an integer between 100 and 150 divisible by 9.   How do i figure this out, I am completely…

A: A number is divisible by 9 if the sum of the digits is divisible by 9.

Q: B. Using induction, prove that 8" -1 is divisible by 7, where ne N

A: (B) 8n-1 is divisible by 7 Now for n=1 81-17=1 For n=2 82-17=64-17=637=9 Now, assume that 8n-1 is…

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Q: Prove that 6-1 is divisible by 5 for any n 21.

A:

Q: How many positive integers greater than 5 and less than 2020 are divisible by 7? are divisible by…

A: 1) Here , 2020/7 = 288.57 So 288 multiples of 7 are there between 5 to 2020 So , 288 integers are…

Q: What is the greatest common divisor of 24 and 44? n If n is a natural number such that gcd(800, n) =…

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Q: Prove that 9? −1 is divisible by 8,for all n ≥ 1

A:

Q: Let p>5 be the prime number.Prove that the expression p4-10p2+9 is divisible by 1920.

A: Given: Let p>5 be the prime number

Q: 8n? a prime number for every n = 1, 2, 3, . ...

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Q: The prime factorization of 7007 is 7.11.13 72.11.13 73.11.13 7.11³.13 o

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Q: 6. What is the sum of all 100 positive odd integers starting with 1? Apply your common sense if it…

A: Given n=100

Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

A:

Q: Determine the possibilities for the final digit of a sixth power of an integer.

A: The last digits of a number are 0,1,2,3,4,5,6,7,8 and 9. The number 0 raised to any number will be 0…

Q: Let p > 5 be the prime number. Prove that the expression p* - 10p? + 9 is divisible by 1920.

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Q: By the mathematical induction, n5 - n - 10 is divisible by 5. O False O True

A:

Q: Show by induction that 21" – 8" is divisible by 13 for all n> 0

A: For the solution follow the next steps.

Q: The Greatest Common Divisor of two prime numbers a and b

A: Prime numbers are those numbers which have only two factors one factor is 1 and other factor is the…

Q: where did 22/7 come from

A: Given ,  A lawn sprinkler is located at the corner of a yard. The sprinkler is set to rotate through…

Q: se mathematical induction to show    n ≥ 0, 7n - 1 is divisible by 6

A: Given: n≥0, 7n-1 To prove: 7n - 1 is divisible by 6

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A: The given equation is,

Q: Show by induction that 19" – 8" is divisible by 11 for all n >0

A:

Q: 27-8^n a prime? Why?

A: answer is in next step

Q: How many of the calculations are prime factorizations of 1260? i. 14 x 30 x 3 ii. 7 x 60 × 3 iii. 42…

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Q: The number of factors of prime numbers are Depends on the prime number 3 0 2 0 None of the mentioned…

A: The number of factors of a prime number is 1 and prime itself. So there are 2 factors of prime…